Metric theory of approximation and geometry of discrete self-similar measures

Abstract

This thesis deals with the theory of metric approximation (also called Diophantine approximation in number theory). More precisely, we prove a very general theorem providing lower-bounds for the Hausdorff dimension of very varied limsup sets - we are also interested in the questions of obtaining upper-bounds of dimensions of such sets. We will apply our results to many problems. For example, we will study the theory of approximation by orbits of systems of iterated functions composed of "overlapping" similarities. We will also give applications of our theorems in number theory and to the multifractal analysis of discrete self-similar measures (i.e. carried by an orbit of the considered iterated function system).

Full text

École Doctorale MSTIC
UMR 8050 - Laboratoire d'Analyse et de Mathématiques
Appliquées
Thèse
Présentée pour l'obtention du grade de DOCTEUR
DE L'UNIVERSITE PARIS-EST
par
Edouard Daviaud
Metric theory of approximation
and geometry of discrete
self-similar measures
Spécialité : Mathematiques
Soutenue le 9 Juin 2023 devant un jury composé de :
Directeur de thèse
Prof. Stéphane Seuret
(Université Paris-Est Créteil)
Directeur de thèse
Prof. Julien Barral
(Université Sorbonne Paris-Nord)
Rapporteur
Prof. Yann Bugeaud
(Université de Strasbourg)
Rapporteur
Prof. Sanju Velani
(University of York)
Examinateur
Prof. Faustin Adiceam
(Université Paris-Est Créteil)
Examinateur
Prof. Céline Esser
(Université de Liège)
Examinateur
Prof. Stéphane Jaard
(Université Paris-Est Créteil)
Examinateur
Prof. Maarit Järvenpää
(University Oulu)

UMR 8050 - Laboratoire d'Analyse et de Mathématiques
Appliquées
Université de Pars-Est Créteil
61, avenue du Général de Gaulle
94010 Créteil
France

Contents
1 Remerciements 1
2 Introduction en français 3
2.1 Théorèmes d'ubiquité . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 Un historique des théorèmes d'ubiquité . . . . . . . . . . 4
2.1.2 Un théorème général d'ubiquité hétérogène . . . . . . . . 7
2.1.3 Etude de l'optimalité du théorème d'ubiquité . . . . . . 9
2.2 Application en approximation diophantienne . . . . . . . . . . . 12
2.3 Cibles rétrécissantes faiblement conformes . . . . . . . . . . . . 14
2.4 Application aux mesures discrètes . . . . . . . . . . . . . . . . . 17
2.5 Conclusion et perspectives . . . . . . . . . . . . . . . . . . . . . 20
3 Introduction 23
3.1 Ubiquity theorems . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.1.1 An historic of ubiquity theorems . . . . . . . . . . . . . . 23
3.1.2 A general ubiquity theorem . . . . . . . . . . . . . . . . 27
3.1.3 Study of optimality in ubiquity theorems . . . . . . . . . 29
3.2 An application in Diophantine approximation . . . . . . . . . . 32
3.3 Weakly conformal shrinking targets . . . . . . . . . . . . . . . . 34
3.4 Application to discrete measure . . . . . . . . . . . . . . . . . . 37
3.5 Conclusion and perspectives . . . . . . . . . . . . . . . . . . . . 40
4 Covering Lemmas, Hausdor measures and IFS 45
4.1 Coveringlemmas .......................... 45
4.2 Hausdor dimension, dimension of measures . . . . . . . . . . . 49
4.2.1 Hausdor measures, dimension and content . . . . . . . 49
4.2.2 Packing measure and packing dimension . . . . . . . . . 52

iv CONTENTS
4.2.3 Dimension of measures . . . . . . . . . . . . . . . . . . . 53
4.2.4 Multifractal analysis of measures . . . . . . . . . . . . . 54
4.3 Iterated function systems . . . . . . . . . . . . . . . . . . . . . . 55
4.3.1 IFS and separation condition . . . . . . . . . . . . . . . 57
4.3.2 Generalities on conformal and weakly conformal IFS . . 60
4.3.3 Weakly conformal systems satisfying AWSC . . . . . . . 67
4.3.4 Dimension properties of self-similar measures . . . . . . . 73
5 Anisotropic ubiquity for quasi-Bernoulli measures 81
5.1 Anisotropique ubiquity and geometric quasi-Bernoulli measures 81
5.2 Proof of Theorem 5.1.5 . . . . . . . . . . . . . . . . . . . . . . . 84
6
µ
-a.c sequences of balls 99
6.1 limsup and
µ
-a.cproperty ..................... 99
6.1.1 Consequences of the
µ
-asymptotic covering property . . . 100
6.1.2 Proof of Theorem 6.1.2 . . . . . . . . . . . . . . . . . . . 103
6.1.3 A version of Borel-Cantelli Lemma . . . . . . . . . . . . 104
6.2 Essential Hausdor content . . . . . . . . . . . . . . . . . . . . 106
6.2.1 Computation of essential content for the Lebesgue measure107
6.2.2 Essential content for weakly conformal measures . . . . . 108
7 Heterogeneous ubiquity theorem 117
7.1 A general heterogeneous ubiquity theorem . . . . . . . . . . . . 117
7.2 Proof of Theorem 7.1.2 and Corollary 7.1.6 . . . . . . . . . . . . 119
7.2.1 Construction of the Cantor set and the measure . . . . . 123
7.2.2 Proof of Corollary 7.1.6 . . . . . . . . . . . . . . . . . . . 133
7.3 Ubiquity theorems and self-conformal measures . . . . . . . . . 135
8 Upper-bound and ubiquity 143
8.1 Weakly redundant sequences of balls . . . . . . . . . . . . . . . 144
8.2 Extraction of
µ
-a.c sub-sequences of balls . . . . . . . . . . . . . 146
8.2.1 Extraction of weakly redundant
µ
-a.c subsequences . . . 148
8.2.2 Extraction of sub-sequences of balls with conditioned mea-
sure .............................149
8.2.3
Gδ
sets of full measure and extraction of
µ
-a.c sub-sequences155

CONTENTS v
8.3 Study of the optimality of ubiquity theorems . . . . . . . . . . . 156
8.3.1 An upper-bound for ubiquity Theorem in the self-similar
case .............................156
8.3.2 Application in the case of balls and rectangles . . . . . . 160
9 Example middle-third Cantor set 163
9.1 Hausdor dimension of
K0
1/3
....................165
9.2 Proof of Theorem 9.0.3 . . . . . . . . . . . . . . . . . . . . . . . 168
10 weakly conformal shrinking targets 171
10.1 Proof of item (1) of Theorem 10.0.1 . . . . . . . . . . . . . . . . 172
10.2 Proof of item (2) of Theorem 10.0.1 . . . . . . . . . . . . . . . . 174
10.3 Proof of item (3) of Theorem 10.0.1 . . . . . . . . . . . . . . . . 175
10.3.1 Variational principle and
C1
weakly conformal IFS . . . 175
10.3.2 Proof of item(3) of Theorem 10.0.1 . . . . . . . . . . . . 179
10.4 case of self-similar IFS . . . . . . . . . . . . . . . . . . . . . . . 181
11 Discrete self-similar measures 183
11.1 Denition and rst result . . . . . . . . . . . . . . . . . . . . . . 183
11.2 Scaling function of discrete self-similar measures . . . . . . . . . 188
11.3 Proof of Proposition 11.1.6 . . . . . . . . . . . . . . . . . . . . . 193
11.4 Multifractal analysis of
χ
under AWSC . . . . . . . . . . . . . . 195
11.4.1 Estimates for
Dχ
for
h≥λc+t
..............198
11.4.2 Multifractal spectrum of
χ
for
h∈[0, λc+t]
.......201

vi CONTENTS

Chapter 1
Remerciements
Je souhaite d'abord remercier chaleureusement Stéphane Seuret et Julien Barral
(mes deux directeurs de thèse) pour les conseils qu'ils m'ont prodigué et la
patience (parfois le courage, il faut bien le dire) dont ils ont fait preuve tout
au long de ma thèse, même après avoir constaté que la 20 ème version de tel
ou tel article changeait du tout au tout pour rendre un résultat un petit peu
plus général. En tout cas, il me semblait inévitable de ne pas énoncer l'adage
suivant qui m'a été répété de très très (très) nombreuses fois:
“
Le mieux est l'ennemi du bien.
”
Je souhaite aussi remercier particulièrement les deux rapporteurs de cette
thèse, Yann Bugeaud et Sanju Velani pour avoir pris le temps de lire, corriger et
commenter mon manuscrit, ainsi que Faustin Adiceam, Céline Esser, Stéphane
Jaard et Maarit Järvenpää pour avoir accepté de faire partie de mon jury.
Je remercie aussi tous les membres du LAMA pour les nombreuses et
intéressantes discussions scientiques et pédagogiques.
Je souhaite également bon courage à Elie, Clément, Nadia (même si elle
n'est plus vraiment doctorante), Irène, Sana, Asma, Teo, Thomas, Houssam et
Valentin pour nir leur thèse. C'est dur mais ça vaut le coup!
Enn, je remercie plus particulièrement mes co-bureau fractalsites Wej-
dene, Quentin, Qian et Guillaume. L'excellent environnement dans lequel nous
avons pu travailler (et pas que travailler) a largement contribué à un certain
épanouissement scientique. Je vous souhaite également bon courage pour nir
votre thèse pour les trois premiers.
Mes amis, Sakina, Magda, Jimmy, Antoine, Charlie, Paul, Antoine, (oui
ça fait deux fois Antoine, mais de nos jours, tout le monde s'appelle Antoine ou
Stéphane alors....), Thimotée, Pierre et David, merci pour vos conseils et votre

2 CHAPTER 1: REMERCIEMENTS
soutien durant ces diciles années de thèses (et plus particulièrement pendant
les trois dernières).
Enn, je dédie aussi cette thèse à ma mère, mon père, ma soeur, César,
Nico, mon frère, Ani, à Papi, à mes oncles et tantes et mes cousins.

Chapter 2
Introduction en français
La théorie de l'approximation métrique cherche à décrire géométriquement
(souvent via le calcul de dimensions de Hausdor) l'ensemble des éléments ap-
proximables par une famille de points
(xn)n∈N
choisie à l'avance dans un espace
métrique
(X, d)
. Historiquement, cette théorie est née des travaux de Dirich-
let portant sur l'approximation des nombres réels par des nombres rationnels
au XIXème siècle. Ces travaux furent complétés et enrichis plus tard, dans
les années 1920, par Jarnik et Besicovitch [47] et récemment, dans les années
2000, par les travaux de Jaard [46], de Bugeaud [15], de Beresnevich et de
Velani [11]. Ce n'est qu'à partir des années 1990, que certains de ces problèmes
d'approximation diophantienne ont été reformulés en terme d'approximation
par des orbites de systèmes dynamiques [42]. Dans ce cadre, on étudie le cas
où la famille
(xn)n∈N
est une orbite, i.e.,
xn=Tn(x)
,
x∈X
, où
X
est un
espace métrique et
T:X→X
est une application mesurable. Pour des exem-
ples où
T
est le doublement de l'angle sur le cercle, une fonction Markovienne
dilatante sur
[0,1]
ou bien pour d'autres examples de fonctions dilatantes, voir
[33,55,65].
Ces ensembles jouent également un rôle important en analyse multifrac-
tale. Par exemple, l'étude des séries de Riemann [46] fait intervenir la vitesse
d'approximation d'un réel par des nombres rationnels. Pour d'autres exem-
ples de travaux mêlant analyse multifractale et approximation diophantienne,
citons également les travaux de Barral, Seuret, Liao, Fan, Shmeling, Troubet-
zkoy, Jaard et Persson [10,33,45,55,64].

4 CHAPTER 2: INTRODUCTION EN FRANÇAIS
2.1 Théorèmes d'ubiquité
2.1.1 Un historique des théorèmes d'ubiquité
Fixons
d∈N∗
et munissons
Rd
de la norme
||(x1, ..., xd)||∞= max1≤i≤d|xi|
(le choix de la norme n'importe en fait pas du tout pour les résultats de ce
manuscrit).
Une façon de donner du sens à la notion d'approximation d'un point
x∈Rd
par une suite de points
(xn)n∈N
de
Rd
et d'en caractériser la vitesse, est
de se donner des suites d'ensembles
(Un(δ))n∈N,
où
δ≥1
(parfois on prend un
multi-paramètre
δ∈[1,+∞)k
, où
k∈N
est xé) vériant:

pour tout
n∈N,
pour tout
δ≥1, xn∈Un(δ),

pour tout
δ≥1,|Un(δ)| → 0,

pour tout
δ′> δ,
pour tout
n∈N, Un(δ′)⊂Un(δ).
On s'intéresse alors pour chaque
δ
, à l'ensemble
Eδ
des points
x
pour lesquels
on peut trouver une innité d'entiers
(nk)k∈N
telle que
x∈Unk(δ).
L'ensemble
de ces points est appelé la limsup
1
des ensembles
(Un(δ))n∈N
et s'écrit
Eδ= lim sup
n→+∞
Un(δ) := \
N∈N[
n≥N
Un(δ).
On dit alors qu'un point
x
est approximable à vitesse
δ
si
x∈Fδ:= Eδ\
Sδ′>δ Eδ′
et on cherche en général à calculer les dimensions de Hausdor (voir
Section 4.2, Lemma 4.2.3 pour une dénition de la dimension de Hausdor)
dimH(Eδ)
et
dimH(Fδ)
(dans les faits, on arrive souvent à calculer
dimH(Fδ)
quand on sait que
Hζ(Eδ)>0
où
ζ
est une jauge appropriée et
Hζ
est la mesure
de Hausdor associée).
Le premier exemple historique de ce genre de calcul est le théorème de
Jarnik-Besicovitch.
Théorème 2.1.1
(Jarnik-Besicovitch [47])
.
Pour
x∈R
et
r > 0,
notons
B(x, r)
la boule fermée centrée en
x
et de rayon
r.
Alors, pour tout
δ≥1,
on
a, notant
Eδ:= lim supq∈N,0≤p≤qB(p
q,1
q2δ)
et
Fδ:= Eδ\Sδ′>δ Eδ′
:
dimH(Eδ) = dimH(Fδ) = 1
δ
(2.1)
1
Notons ici que la dénition de l'ensemble
lim supn→+∞Un
est indépendante de la numéro-
tation choisie pour les ensembles
{Un}n∈N,
c'est à dire que pour toute bijection
ϕ:N→N,
on
a
lim supn→+∞Un= lim supn→+∞Uϕ(n).
En particulier, nous prendrons dans ce manuscrit
parfois des limsups d'ensembles sans préciser explicitement la numérotation choisie.

2.1 THÉORÈMES D'UBIQUITÉ 5
Mentionnons aussi d'autres travaux portant sut l'approximation rationnelle
[11,15,16,27] (approximation rationnelle homogène et inhomogène dans
Rd
)
et [1,17,18,54] (approximation rationnelle d'éléments du Cantor triadique).
Usuellement, pour calculer les dimensions de ces ensembles, on procède
séparément pour la minoration et la majoration. Dans la pratique, majorer
dimH(Eδ)
s'avère souvent ou relativement simple ou très complexe (le lecteur
notera que dans ce manuscrit, toutes les preuves des majorations sont relative-
ment plus faciles que celle des minorations de
dimH(Eδ)
).
Les théorèmes d'ubiquité, plus souvent appelés principes de transfert de
masse ([11]), sont des outils établis pour traiter de la minoration de
dimH(Eδ).
Ces théorèmes fonctionnent de la façon suivante: on cherche une mesure
µ∈ M(Rd) := 
mesures de probabilité sur
Rd
(qu'on espère adaptée au
problème) de sorte que
µ(E1)=1.
Ensuite, en utilisant les propriétés géométriques
de la mesure
µ
(auto-similarité, analyse multifractale par exemple), on cherche
à établir une minoration de
dimHEδ
pour tout
δ > 1.
Voici un exemple impor-
tant de ce genre de théorème. On dit qu'une mesure
µ∈ M(Rd)
est
s
-Alfhors
régulière, où
0≤s≤d,
s'il existe deux constantes
0< C1≤C2
de sorte que,
pour tout boule
B(x, r)
avec
x∈supp(µ) :=
le support topologique de
µ
et
0< r ≤1,
on a
C1rs≤µ(B(x, r)) ≤C2rs.
Un exemple de telle mesure est la mesure de Lebesgue
Ld
sur
[0,1]d.
Théorème 2.1.2
(Beresnevich-Velani [11])
.
Soit
0≤s≤d
et
µ∈ M(Rd)
une
mesure
s
-Alfhors régulière. Soit
(Bn)n∈N
une suite de boules fermées centrées
sur
supp(µ)
et satisfaisant:

|Bn| → 0,

µ(lim supn→+∞Bn)=1.
Soit
δ > 1,
et
B
une boule centrée sur
supp(µ).
Notons pour tout
s0≥0
,
Hs0
la mesure de Hausdor de dimension
s0
(voir
(4.2.2)
) et pour
n∈N
,
Bδ
n=B(xn, rδ
n)
si
Bn=B(xn, rn).
On a
Hs
δ(lim sup
n→+∞
Bδ
n∩B)=+∞.
En particulier, on retrouve un résultat établi par Jaard (1999),
dimH(lim sup
n→+∞
Bδ
n)≥s
δ.
Dans le cas où
µ=Ld
, on peut aussi traiter le cas plus général où au

6 CHAPTER 2: INTRODUCTION EN FRANÇAIS
lieu de déterminer une minoration de
lim supn→+∞Bδ
n
pour tout
δ > 1,
on
décrit une quantité géométrique, le contenu de Hausdor (voir Section 4.2,
equation (4.7)), à estimer pour minorer (ecacement)
dimH(lim supn→+∞Un)
pour n'importe quelle suite d'ouverts
(Un)n∈N
satisfaisant
Un⊂Bn.
Théorème 2.1.3
(Koivusalo-Rams [51])
.
Soit
(Bn)n→+∞
une suite de boules
de
[0,1]d
vériant
|Bn| → 0
,
(Un)n∈N
une suite d'ouverts vériant pour tout
n∈N, Un⊂Bn
et
0≤s≤d.
Supposons que:

la suite
(Bn)n∈N
vérie
Ld(lim sup
n→+∞
Bn) = 1,

pour tout
n∈N
assez grand
Hs
∞(Un)≥ Ld(Bn).
Alors
dimH(lim sup
n→+∞
Un)≥s.
Ajoutons ici que des études de dimensions de Hausdor d'ensemble limsup
générés par des ensembles aléatoires faisaient déjà apparaître explicitement le
rôle du contenu de Hausdor pour caractériser la dimension de l'ensemble lim-
sup [28,38,48,49] et que récemment, d'autres résultats d'ubiquité basés sur le
contenu de Hausdor ont également été établis [29]. Un exemple d'application
est le cas où chaque
Un
est un rectangle ouvert. En eet, xons
(τ1, ..., τd)∈
[1,+∞)d
. Soit
(Bn:= xn+Qd
i=1[−rn, rn])n∈N
une suite de boules de
[0,1]d
vériant
|Bn| → 0
et
Ld(lim supn→+∞Bn)=1.
Posons
Rn=xn+
d
Y
i=1
(−rτi
n, rτi
n).
(2.2)
Alors, pour tout
0≤s≤d,
le contenu de Hausdor de dimension
s
des
rectangles
Rn
peut être estimé explicitement en fonction de
rn
et des nombres
τ1, ..., τd
à une constante multiplicative près
C
qui ne dépend que de
d
. En
particulier, on obtient, que pour
s0= min1≤i≤d
d+P1≤i≤dτk−τi
τk
([51]),
Hs0
∞(Rn)≥Crd
n≥CLd(Bn).
On vérie facilement que ceci implique que pour tout
s < s0
et
n
assez grand,
Hs
∞(Rn)≥rd
n=Ld(Bn).

2.1 THÉORÈMES D'UBIQUITÉ 7
Par application du Théorème 2.1.3, on obtient un résultat qui avait été étblit
par Wang, Wu et Xu dans [75],
dimH(lim sup
n→+∞
Rn)≥s0= min
1≤i≤d
d+P1≤i≤dτk−τi
τk
.
Lorsque la mesure n'est pas homogène (Alfhors régulière), le problème est plus
complexe.
Le théorème de Barral-Seuret traite le cas où la mesure est multinômiale
de support
[0,1]d
(mais on en déduit relativement simplement le cas auto-
similaire satisfaisant l'Open Set Condition qu'on raccourcira en OSC, voir Sec-
tion 4.3, Dénition 4.3.3 pour plus de détails sur les diérentes conditions de
séparation). Pour
µ
une mesure auto-similaire, notons
dim(µ)
dimension de
µ
(voir Section 4.2, Dénition 4.2.12 et Section 4.3, Théorème 4.3.15).
Théorème 2.1.4
(Barral-Seuret [8])
.
Soit
µ∈Rd
une mesure multinômiale
de support
[0,1]d
,
(Bn)n∈N
une suite de boules de
[0,1]d
vériant
|Bn| → 0
.
Si
µ(lim sup
n→+∞
Bn)=1,
alors, pour tout
δ≥1,
dimH(lim sup
n→+∞
) = dim(µ)
δ.
(2.3)
Dans ce manuscrit, on développe un outil qui joue le même rôle que le
contenu de Hausdor pour le Théorème 2.1.3 mais dans le cas où la mesure
µ∈ M(Rd)
est quelconque. Nous montrerons aussi que le théorème d'ubiquité
résultant étend à la fois les Théorèmes 2.1.3 et 2.1.4.
2.1.2 Un théorème général d'ubiquité hétérogène
Commençons par introduire la quantité géométrique suivante.
Dénition 2.1.5
(D. [25])
.
Soit
µ∈ M(Rd) 0 ≤s≤d
et
A⊂Rd
un ensemble
mesurable. Le contenu de Hausdor
µ
-essentiel
s
-dimensionnel de
A
est déni
par
Hµ,s
∞(A) = inf {Hs
∞(E) : E⊂A, µ(E) = µ(A)}.
(2.4)
Pour
µ∈ M(Rd)
, notons
dimH(µ)
la dimension de Hausdor inférieure de
µ
(dénie en Section 4.2, Dénition 4.2.12). Le théorème d'ubiquité associé à
cette quantité géométrique est le suivant (il s'agit en fait d'une version simpliée
du Théorème 7.1.2).

8 CHAPTER 2: INTRODUCTION EN FRANÇAIS
Théorème 2.1.6
(D. [25])
.
Soit
µ∈ M(Rd)
,
(Bn)n∈N
une suite de boules
satisfaisant
|Bn| → 0,(Un)n∈N
une suite d'ouverts satisfaisant, pour tout
n∈N
,
Un⊂Bn
et
0≤s < dimH(µ).
Supposons que

µ(lim supn→+∞1
2Bn)=1,

pour
n
assez grand,
Hµ,s
∞(Un)≥µ(Bn).
Alors on a
dimH(lim sup
n→+∞
Un)≥min {s, dimH(µ)}.
(2.5)
Dans le Chapitre 7, le résultat que nous montrerons est en réalité un peu plus
général que le Théorème 2.1.6.
Evidemment, pour pouvoir appliquer ce théorème à une mesure
µ
, il
faut comprendre cette mesure susamment nement et être capable d'estimer
Hµ,s
∞(Un)
pour tout
0≤s≤dimH(µ)
et chacun des ouverts
Un
. Il se trouve
que lorsque la mesure est faiblement conforme (Section 4.3, Denition 4.3.1),
on peut estimer précisément le contenu
µ
-essentiel de n'importe quel ouvert
sans aucune hypothèse de séparation sur l'IFS
.
Théorème 2.1.7
(D. [23])
.
Soit
µ∈ M(Rd)
une mesure faiblement conforme
de support
K
.
Pour tout
0≤s < dim(µ)
, pour tout
ε > 0
, il existe une constante
c=c(d, µ, s, ε)>0
qui dépend de
d
,
µ
,
s
et
ε
, telle que pour toute boule fermée
B=B(x, r)
centrée sur
K
et
r≤1
, pour tout ouvert
Ω
, on a
c(d, µ, s, ε)|B|s+ε≤ Hµ,s
∞(˚
B)≤ Hµ,s
∞(B)≤ |B|s
c(d, µ, s, ε)Hs+ε
∞(Ω ∩K)≤ Hµ,s
∞(Ω) ≤ Hs
∞(Ω ∩K).
(2.6)
Pour tout
s > dim(µ)
,
Hµ,s
∞(Ω) = 0.
Ainsi la combinaison des Théorèmes 2.1.6 et 2.1.7 permet de retrouver le
Théorème 2.1.3 en prenant
µ=Ld
(dans ce cas,
K= [0,1]d
) et de montrer les
corollaires suivants.
Corollaire 2.1.8.
Soit
µ∈ M(Rd)
une mesure faiblement conforme et
(Bn)n∈N
une suite de boules centrées sur
supp(µ)
vériant
|Bn| → 0.
Si
µ(lim sup
n→+∞
Bn)=1,

2.1 THÉORÈMES D'UBIQUITÉ 9
alors, pour tout
δ≥1,
dimH(lim sup
n→+∞
Bδ
n)≥dim(µ)
δ.
Corollaire 2.1.9.
Soit
µ∈ M(Rd)
une mesure faiblement conforme de sup-
port
[0,1]d
et
(Bn)n∈N
une suite de boules de
[0,1]d
vériant
|Bn| → 0.
Soit
(τ1, ..., τd)∈[1,+∞)d
et
Rn
, déni par
(2.2)
.
Si
µ(lim sup
n→+∞
Bn)=1,
alors
dimH(lim sup
n→+∞
Rn)≥min
1≤i≤d
dim(µ) + P1≤i≤dτk−τi
τk
.
Notons que le Corollaire 2.1.8 étend directement le Théorème 2.1.4.
Mentionnons aussi que pour tout
1≤τ1≤... ≤τd
, toute suite de matri-
ces de rotations
(Ok,n)k∈N,0≤n≤2k−1∈ Od(R)N,
presque pour toute suite de réal-
isations de variables aléatoires i.i.d. et uniformément distribuées dans
[0,1]d
,
(xk,n)k∈N,0≤n≤2k−1
, il est démontré dans [48] (et la preuve utilise implicitement
le Corollaire 2.1.9 dans le cas de la masure de Lebesgue) que la famille de
rectangles
R=(xk,n +Ok,n
d
Y
i=1
(−2−kτi,2kτi),0≤n≤2k−1, k ∈N)
vérie
dimH(lim sup
R∈R
R) = min
1≤k≤d(d+Pk
j=1 τk−τj
τk).
En particulier la borne donnée par le Corollaire 2.1.9 est optimale. Nous
verrons un peu plus tard qu'elle l'est même en un sens plus fort.
Les Corollaires 2.1.8 et 2.1.9 illustrent l'intérêt du Théorème 2.1.6: pour
minorer la quantité
dimH(lim supn→+∞Un)
, on établit pour tout
n∈N
une
relation qui ne dépend que de
Bn
,
Un
et
µ
.
Dans le Chapitre 8, on étudie l'optimalité des bornes données par le
Théorème 2.1.6.
2.1.3 Etude de l'optimalité du théorème d'ubiquité
Les minorations établies par les Théorèmes 2.1.3,2.1.4 et 2.1.6 n'ont d'intérêt
que si elles sont, en un sens satisfaisant, optimales. Une façon de traiter la

10 CHAPTER 2: INTRODUCTION EN FRANÇAIS
question de l'optimalité de ces bornes est de montrer que si l'on ne dispose pas
d'informations supplémentaires (par rapport à celles données par les hypothèses
de ces théorèmes) sur la suite de boules
(Bn)n∈N
, il n'y a pas de raison que
dimH(lim supn→+∞Un)
soit supérieur à la borne donnée. Une méthode pour
établir que c'est le cas est d'extraire de
(Bn)n∈N
, quand c'est possible, une sous-
suite
(Bϕ(n))n∈N
qui vérie toujours les hypothèses des théorèmes d'ubiquité
et telle que
dimH(lim supn→+∞Uϕ(n))
est précisément la borne donnée par le
théorème. Dans cette situation, nous sommes assurés que l'on ne peut pas
obtenir de meilleure borne sous les mêmes hypothèses.
Introduisons la denition suivante, qui est une variante (équivalente) de
la notion de suite de boules faiblements redondantes introduite par Barral et
Seuret et rappelée également plus tard dans ce manuscrit (Dénition 8.1.1) .
Denition 2.1.1.
Soit
B= (Bn)n∈N
une suite de boules vériant
|Bn| → 0.
Pour
k∈N
, dénissons
Tk(B) = Bn∈ B : 2−k−1≤ |Bn|<2−k
(2.7)
Nk(B) = max
B∈Tk(B)#{B′∈ Tk(B) : B∩B′=∅}.
On dit que
B
est faiblement redondante si
lim
k→+∞
log Nk(B)
k= 0.
Heuristiquement, une suite de boules
(Bn)n∈N
est faiblement redondante
si pour chaque
k∈N
, chaque boule
Bn
satisfaisant
|Bn| ≈ 2−k
intersecte moins
de
2o(k)
autres boules
Bn′
satisfaisant aussi
|Bn′| ≈ 2−k.
Une remarque impor-
tante est que la faible redondance est une hypothèse naturelle pour étudier
l'optimalité des théorèmes d'ubiquité et qu'elle porte uniquement sur la famille
de boules (il n'y pas d'hypothèse impliquant une mesure). La conséquence
techniquement importante de cette propriété est que, pour tout mesure
µ∈
M([0,1]d)
et tout
ε > 0,
on a
X
n∈N|Bn|εµ(Bn)<+∞.
(2.8)
Beaucoup de suites de boules sont faiblement redondantes, comme la suite
des boules rationnelles
ÄB(p
q,1
q2)äq∈N,0≤p≤q,q∧p=1
, mais il existe aussi des exem-
ples de suites naturelles de boules ne satisfaisant pas cette hypothèse. C'est le
cas par example de la suite
ÄB(p
q,1
q2)äq∈N,0≤p≤q.
Cependant, on peut montrer
pour certains de ces exemples que, bien qu'elles ne satisfassent pas l'hypothèse
de faible redondance, ces suites vérient tout de même l'équation (2.8) pour une

2.1 THÉORÈMES D'UBIQUITÉ 11
mesure naturelle (
ÄB(p
q,1
q2)äq∈N,0≤p≤q
pour la mesure de Lebesgue). Ajoutons
aussi que d'autres auteurs utilisent des hypothèses comparables mais lègère-
ments diérentes, telles que l'hypothèse de régularité optimale d'un système de
points approximants (voir [15]).
Dans le cas où la mesure est faiblement conforme, l'hypothèse de faible
redondance permet d'établir le théorème de majoration suivant.
Théorème 2.1.10
(D.[24])
.
Soit
µ∈ M(Rd)
une mesure faiblement conforme,
(Bn)n∈N
une suite de boules faiblement redondante, centrées sur
supp(µ)
et
satisfaisant
|Bn| → 0,(Un)n∈N
une suite d'ouverts vériant
Un⊂Bn
et
0≤
s < dim(µ).
Si pour tout
n
assez grand,
Hµ,s
∞(Un)≤µ(Bn),
alors on a
dimH(lim sup
n→+∞
Un)≤s.
Notons qu'étant donnés
µ∈ M(Rd)
,
(Bn)n∈N
et
(Un)n∈N
satisfaisant pour
tout
n∈N
,
Un⊂Bn
, il n'est pas vrai en général qu'il existe un exposant
critique
s0
de sorte que pour tout
s<s0
, pour
n
assez grand,
Hµ,s
∞(Un)≥µ(Bn)
et pour tout
s>s0,
pour
n
assez grand
Hµ,s
∞(Un)≤µ(Bn).
Néanmoins, dans le cas des mesures faiblements conformes, lorsque le
choix des ensembles
Un
est fait de façon cohérente (par exemple, si l'on prend
des rectangles contractés via les mêmes facteurs de contraction
(τ1, ..., τd)∈
[1,+∞)d
ou bien des boules
Bδ
n
,
δ≥1
étant xé à l'avance), le Théorème
2.1.10 permet de conclure à l'optimalité des bornes données par les Corollaires
2.1.8 et 2.1.9.
Corollaire 2.1.11.
Soit
µ∈ M(Rd)
une mesure faiblement conforme et
(Bn)n∈N
une suite de boules centrées sur
supp(µ)
vériant
µ(lim supn→+∞1
2Bn) = 1
et
|Bn| → 0
. Alors:
1. Il existe une sous-suite
(Bϕ(n))n∈N
vériant
µ(lim supn→+∞Bϕ(n)) = 1
et,
pour tout
δ≥1,
dimH(lim sup
n→+∞
Bδ
ϕ(n)) = dim(µ)
δ.
2. Si de plus
supp(µ) = [0,1]d
, alors il existe une sous-suite
(Bϕ(n))n∈N
véri-
ant
µ(lim supn→+∞Bϕ(n)) = 1
et, pour tout
(τ1, ..., τd)∈[1,+∞)d,
dimH(lim sup
n→+∞
Rϕ(n)) = min
1≤i≤d
dim(µ) + P1≤i≤kτk−τi
τk
,

12 CHAPTER 2: INTRODUCTION EN FRANÇAIS
où
Rn
est donné par
(2.2)
.
En particulier, les bornes données par les Corollaires 2.1.8 et 2.1.9 sont
optimales en un sens relativement satisfaisant: sous des hyptohèses raisonables
sur la suite
(Bn)n∈N,
on peut toujours extraire une sous-suite
(Bϕ(n))n∈N
qui
vérie encore
µ(lim supn→+∞Bϕ(n))=1
et telle que la limsup générée par les
ensembles contractés associés (boules ou rectangles) ont les dimensions atten-
dues pour tous les ratios de contractions.
La prochaine section présente une application en approximation diophanti-
enne. Ces résultats sont établis dans le Chapitre 9.
2.2 Application en approximation diophantienne
Comme mentionné précédemment, une des problématiques classiques en ap-
proximation diophantienne est de déterminer les dimensions de divers ensem-
bles de points approximables par des nombres rationnels. Considérons
f0:
R→R, f1:R→R
et
f2:R→R
dénies par
f0:x→1
3x
,
f1(x) = 1
3x+1
3
et
f2(x) = 2
3+1
3x
et notons
K1/3
l'attracteur (voir Section 4.3, Proposition 4.3.1)
de
{f0, f2}
. Un problème posé par Mahler consiste à déterminer la dimension
de l'ensemble
K1/3∩Fδ
(ou
Fδ
est déni dans le Théorème 2.1.1) des points
du Cantor triadique
K1/3
qui sont approximables par des rationnels à vitesse
δ≥1
xée. Ce problème reste non résolu encore aujourd'hui mais les théorèmes
d'ubiquité ont tout de même permis d'établir des résultats partiels.
Théorème 2.2.1
(Beresnevich-Velani [11], Barral-Seuret [8])
.
Pour tout
δ≥1,
dimHÇlim sup
0≤p≤q,q→+∞
B(p
q,1
q2δ)∩K1/3å≥dimH(K1/3)
2δ.
La valeur de
dimHÄlim sup0≤p≤q,q→+∞B(p
q,1
q2δ)∩K1/3ä,
où
δ≥1,
a don-
née lieu à plusieurs conjectures
(1) Levesley, Salp, Velani [54]:
dimHÇlim sup
0≤p≤q,q→+∞
B(p
q,1
q2δ)∩K1/3å=dimH(K1/3)
δ.
(2) Bugeaud-Durand [18]:
dimHÇlim sup
0≤p≤q,q→+∞
B(p
q,1
q2δ)∩K1/3å= max ßdimH(K1/3)
δ,2
δ+ dimH(K1/3)−1™.

2.2 APPLICATION EN APPROXIMATION DIOPHANTIENNE 13
Grâce au Théorème 2.1.3 (qui est impliqué par Théorème 2.1.6), on peut obtenir
une caractérisation complète des points approximables par des rationnels dans
un ensemble qui contient
K1/3
et qui a même dimension.
Notons
Λ = {0,1,2}
et le shift sur
ΛN
,
σ
, déni par
σ((i1, i2, ...)) = (i2, ...)
.
La projection canonique de
ΛN
sur
[0,1]
est l'application
π:x= (xn)n∈N7→ lim
n→+∞fx1◦... ◦fxn(0).
(2.9)
Dénition 2.2.2.
Soit
ϕ: ΛN→ {0,1}
dénie par



ϕ(x) = 1
si
x1= 1
ϕ(x) = 0
si
x1= 0
ou
2
et
K(0)
1/3=πÅßx∈ΛN: lim inf
k→+∞
Skϕ(x)
k= 0™ã,
où
(Skϕ)k∈N
est la somme de Birkho de
ϕ
dénie pour
x∈ΛN
par
Sk(ϕ)(x) = 1
k
k−1
X
i=0
ϕ(σi(x)).
Il est démontré dans ce manuscrit (Proposition 9.1.5) que
dimHK(0)
1/3=log 2
log 3(= dimHK1/3).
On a le résultat suivant:
Théorème 2.2.3
(D.[25])
.
Pour tout
δ≥1,
dimHlim sup
0≤p≤q,q→+∞
B(p
q,1
q2δ)∩K(0)
1/3= min ßlog 2
log 3,1
δ™.
(2.10)
On observe un phénomène de saturation: pour
1≤δ≤log 3
log 2
, la dimen-
sion
dimHlim sup0≤p≤q,q→+∞B(p
q,1
q2δ)∩K(0)
1/3
est constante égale à
log 2
log 3
, puis
décroit en
1
δ
lorsque
δ≥log 3
log 2 .
Dans la prochaine section, nous présentons une application du Corollaire
2.1.8 aux problèmes de cibles rétrécissantes faiblements conformes.

14 CHAPTER 2: INTRODUCTION EN FRANÇAIS
2.3 Cibles rétrécissantes faiblement conformes
Soit
m≥2
et
S={f1, ..., fm}
un système faiblement conforme et notons
Λ = {1, .., m}
,
Λ∗=Sk≥0Λk
et
K
son attracteur. Etant donné
x∈K,
on
s'intéresse à l'ensemble des éléments approximables par l'orbite de
x
sous
S
,
c'est-à-dire, aux ensembles de la forme
W(x, ψ) = lim sup
i∈Λ∗
B(fi(x), ψ(i)),
où
ψ: Λ∗→R
vérie
lim|i|→+∞ψ(i)=0
et pour
i= (i1, ..., ik), fi=fi1◦...◦fik.
Historiquement, l'étude de ces ensembles est reliée à celle des cibles rétré-
cissantes sur l'attracteur d'un système dynamique répulsif [42].
Plus précisément, supposons pour simplier que l'IFS
S
soit auto-similaire
et satisfasse pour tout
1≤i=j≤m,
Im
(fi)∩
Im(
fj) = ∅
. Alors son attracteur
K
peut être vu comme l'attracteur associé à la fonction mesurable répulsive
F
dénie par
F(x) = f−1
i(x)
si
x∈
Im
(fi).
Soit
δ≥1
et
ψ: Λ∗→Λ∗
la fonction
dénie par
ψ(i) = cie−nδ.
L'ensemble
W(x, ψ)
se réécrit simplement comme
W(x, ψ) = z:Fn(z)∈B(x, e−nδ )
pour une innité d'entiers
n∈N,
(2.11)
qui sont justement les ensembles introduits et étudiés par Hill et Velani dans
[42].
Il existe cependant a priori plusieurs façons naturelles de choisir les classes
de fonctions
ψ
que l'on considère:

D'un point de vue dynamique, il est logique de considérer des fonctions
ψ
de la forme
ψ(i) = |fi(K)|δ
, où
δ≥1,
puisqu'on a naturellement,
lim sup
i∈Λ∗
B(fi(x),|fi(K)|) = K.
Dans ce cas, le Théorème 2.1.6 permet de montrer le résultat suivant:
Théorème 2.3.1
(D. [23])
.
Notons
dim(S)
la dimension de conformalité
de l'IFS
S
(Dénition
(4.3.19)
). Supposons que
dimH(K) = dim(S)
.
Alors pour tout
δ≥1
, pour tout
x∈K
, on a
dimHÇlim sup
i∈Λ∗
B(fi(x),|fi(K)|δ)å=dimH(K)
δ.
(2.12)

En approximation diophantienne, on préfère prendre des fonctions
ψ
de
la forme
ψ(i) = |fi(K)|g(|i|)
où
g:N→R
est une fonction décroissante

2.3 CIBLES RÉTRÉCISSANTES FAIBLEMENT CONFORMES 15
(pour l'étude des ensemble
W(x, ψ)
dans ce cadre sous l'hypothèse que
le système vérie l'OSC, citons les travaux de Hill et Velani et de Allen
et Bárány [2,42]).
Cette diérence de point de vue est justiée par (2.11) et provient égale-
ment du fait que l'on cherche à faire des analogies entre l'étude de ces
ensembles et le théorème de Khintchine dans le cadre rationnel que l'on
rappelle. Notons
ϕ:N→N
la fonction d'Euler, dénie par
ϕ(q) = # {1≤p≤q:p∧q= 1}.
Théorème 2.3.2
(Khintchine [50])
.
Soit
ψ:N:R+
une fonction telle
que
q→ψ(q)
q
soit monotone. Alors
LdÇlim sup
q∈N,0≤p≤q,p∧q=1
Bp
q,ψ(q)
qå= 1 ⇔X
q≥1
ϕ(q)ψ(q)
q= +∞.
Ajoutons qu'il était conjecturé par Dun et Schaeer que le Théorème
2.3.2 restait valide sans l'hypothèse de monotonie de
q→ψ(q)
q
et que cette
conjecture a été résolue récemment par Maynard et Koukoulopoulos [52].
Pour rendre cette analogie plus concrète, supposons pour simplier que
le système
S
soit auto-similaire et appelons
0< c1, ..., cm<1
les ratios
de contraction des fonctions
f1, ..., fm.
Rappelons que dans le cas auto-
similaire,
dim(S)
est le réel solution de
X
1≤i≤m
cdim(S)
i= 1.
On dispose toujours sur le codage
ΛN
d'une mesure naturelle
ν
dénie
pour tout
i= (i1, ..., ik)∈Λ∗
, par
ν(i) = cdim(S)
i.
Soit
µ=ν(π−1)
, où
π
est la projection canonique de
ΛN
sur
K
(dénie
ci-dessous en (2.9)). L'idée est d'obtenir des résultats de dichotomies de
la forme suivante (où
s≥0
est non spécié ici):



si
Pi∈Λ∗ψ(i)s= +∞,
alors
µ(W(x, ψ)) = 1
si
Pi∈Λ∗ψ(i)s<+∞,
alors
µ(W(x, ψ)) = 0.
Ceci en tête, il est clair que la fonction
ψ
ne peut pas être arbitraire et
que
ψ(i)
doit dépendre de la génération du mot
i
, si on espère recouvrir

16 CHAPTER 2: INTRODUCTION EN FRANÇAIS
un ensemble de large mesure par des boules
{B(fi(x), ψ(i))}i∈Λn
, pour
une innité d'entiers
n∈N.
Pour des exemples de tels théorèmes dans le cadre auto-similaire, référons à
[54] et mentionnons aussi le théorème suivant:
Théorème 2.3.3
(Baker [3])
.
Supposons que le système
S
satisfasse l'une des
conditions suivantes


−Pm
i=1 cdim(S)
ilog(cdim(S)
i)<−2 log(Pi≤i≤mc2 dim(S)
i)
c1=... =cm.
Soit
ψ: Λ∗→R
de la forme
ψ(i) = |fi(K)|g(|i|)
, où
g:N→R
est positive et
décroissante. Si
X
n≥0
ng(n)dim(S)= +∞,
alors, pour tout
x∈K,
µ(W(x, ψ)) = 1.
Le Théorème 2.3.3 combiné au Théorème 2.1.6 nous permet de prouver
le résultat suivant:
Théorème 2.3.4
(D. [23])
.
Soit
g:N→(0,+∞)
une fonction positive décrois-
sante. Posons
sg= inf 

s≥0 : X
k≥0X
i∈Λk
k(|fi(K))|g(k))s<+∞

.
(2.13)
Alors, sous les hypothèses du Théorème 2.3.3, si de plus
dim(µ) = dim(S),
pour tout
δ≥1,
on a
dimHlim sup
i∈Λ∗
B(fi(x),(|fi(K)|g(|i|))
δsg
dim(S))=dim(S)
δ.
(2.14)
Remarquons que (2.14) donne une caractérisation complète de la dimen-
sion des ensembles
W(x, ψ)
lorsque
ψ
est de la forme
ψ(i)=(|fi(K)|g(|i|))s
,
pour tout
s≥0
et
g:N→N
positive décroissante.
La prochaine section présente une application en analyse multifractale de
mesures discrètes, développée au Chapitre 11.

2.4 APPLICATION AUX MESURES DISCRÈTES 17
2.4 Application à l'analyse multifractale de mesures
discrètes
Soit
m≥2
et
S={f1, ..., fm}
un système de similitudes contractantes de
rapports de contraction
0< c1, ..., cm<1
et soit
K
l'attracteur de
S
. Etant
donné
(q1, ..., qm+1)∈(0,1)m+1
un vecteur de probabilité, nous étudions les
mesures de probabilité solutions de l'équation perturbée
χ(·) = X
1≤i≤m
qiχ(f−1
i(·)) + qm+1δx0,
(2.15)
où
δx
désigne la mesure de Dirac au point
x.
Ces mesures ont été introduites
par Snigireva et Olsen dans [70] sous l'appellation de mesures auto-similaires
in-homogènes de mesure de condensation
δx0
.
On peut montrer (cf Proposition 11.1.1) que de telles mesures sont dis-
crètes et sont de la forme
χ(·) = CX
i∈Λ∗
pict
iδfi(x0),
(2.16)
où
t > 0,(p1, ..., pm)
est un autre vecteur de probabilité,
C > 0
est une constante
de renormalisation an que
χ
soit une mesure de probabilité.
Des mesures très similaires ont été obtenues par Mandelbrot et Riedi
[56] en considérant des mesures réelles obtenues en inversant les fonctions de
reparatitions associées à des mesures auto-similaires satisfaisant l'OSC et par
Barral et Seuret en considerant le cas d'une mesure de Gibbs sur un Cookie-
cutter dans [9]. Les propriétés multifractales de ces mesures ont été étudiées
par Falconer dans [31], par Barral et Seuret dans [7,9] et par Mandelbrot
et Riedi dans [56]. Ces derniers, en ne considérant seulement des dimensions
locales obtenues via des limites exactes, Mandelbrot et Riedi n'ont pas relevé
un phénomène intéressant mis en lumière par Falconer, Barral et Seuret.
L'analyse ne des propriétés multifractales de mesures est un sujet qui a
vu le jour dans les années 1970 avec les travaux de Mandelbrot [57,58] et de
Olsen [6062]. Cette étude consiste, étant donnée
µ∈ M(Rd)
, à déterminer
d'une part le spectre des dimensions des ensembles suivants (voir Section 4.2,
Dénition 4.2.12)
Eh
µ={x∈supp(µ) : dim(µ, x) = h},
‹
Eh
µ=x∈supp(µ) : dim(µ, x) = dim(µ, x) = h,
(2.17)

18 CHAPTER 2: INTRODUCTION EN FRANÇAIS
c'est-à-dire à déterminer pour tout
h∈R,



Dµ(h) = dimH(Eh
µ)
et
‹
Dµ(h) = dimH(‹
Eh
µ).
On cherche généralement à comparer les dimensions de ces ensembles à la
transformée de Legendre d'une fonction d'échelle, notée
τµ
et dénie pour tout
q∈R
par
τµ(q) = lim inf
n→+∞
−log PD∈Dn:µ(D)=0 µ(D)q
n
ou
Dn
dénote l'ensemble des cubes dyadiques de génération
n
.
Fixons
µ
la mesure auto-similaire associée à
S
et
(p1, ..., pm)
, c'est-à-dire
la mesure solution de l'équation
µ(·) =
m
X
i=1
piµ(f−1
i(·)).
Nous allons voir que l'analyse multifractale de la mesure
χ
, donnée par
(2.16), dépend à la fois de l'analyse multifractale de la mesure auto-similaire
µ
et
de la vitesse d'approximation d'un point
x
par les points de l'orbite
(fi(x0))i∈Λ∗.
Introduisons la dénition suivante.
Dénition 2.4.1.
Soit
x∈K
. Le degré d'approximation de
x
par rapport à
(fi(x0))i∈Λ∗
, noté
∆x
, est déni par
∆x= sup ß∆ : x∈lim sup
n→+∞
B(fi(x0),(2ci)∆)™.
Dans le cas où le point de base de (2.16)
x0
n'appartient pas à l'attracteur
K
de
S
, tous les points de
supp(χ)
, excepté
{fi(x0)}i∈Λ∗
, qui est de dimension
0
, sont mal approchés par ces points, i.e, ont pour degré
1
. Dans ce cas l'analyse
multifractale de
χ
se déduit facilement de celle de
µ.
Théorème 2.4.2
(D.[26])
.
Soit
x0/∈K.
Alors, pour
t > 0
, la mesure
χ
dénie
par
(2.16)
existe. De plus, pour tout
h≥0,
Dχ(h) = Dµ(h−t).
(2.18)
Remarquons que ce résultat est vrai sans hypothèse de séparation sur la
mesure
µ
. Une conséquence directe de ce résultat est que, lorsque l'attracteur
K
a mesure de Lebesgue
0
, pour Lebesgue presque tout
x0,
la mesure
χ
a
pour spectre multifractal un translaté du spectre de la mesure
µ.
En d'autres
termes, pour simuler une mesure qui a pour spectre multifractal le spectre de

2.4 APPLICATION AUX MESURES DISCRÈTES 19
la mesure
µ
translaté de
t
, il sut de tirer aléatoirement un point
x0
suivant la
loi uniforme et de simuler la mesure
χ
associée. Ajoutons aussi que l'on montre
dans ce manuscript que
χ
ne satisfait pas le formalisme multifractal même si
µ
le satisfait (car
τχ(q)=0
pour
q≥1
).
Lorsque
x0
est choisi sur
K
, le Corollaire 2.1.8 permet, sous certaines
hypothèses de calculer le spectre de la mesure
χ
associée.
Théorème 2.4.3
(D. [26])
.
Soit
m≥2
,
S={f1, ..., fm}
IFS auto-similaire
satisfaisant AWSC (Denition 4.3.3) sans overlaps exacts et
K⊂Rd
l'attracteur
de
S
. Soit
(p1, ..., pm)
un vecteur de probabilité,
µ
la mesure auto-similaire as-
sociée avec
(p1, ..., pm), x0∈K
et
C > 0
la constante de normalisation telle
que la mesure
χ(·) = CX
i∈Λ∗
ct
ipiδfi(x0),
soit une mesure de probabilité.
Posons
qc= min {q:τµ(q) + qt = 0}
et
λc=τ′
µ(qc)
. Alors:
1. La fonction
τχ
satisfait









pour tout
q≥qc, τχ(q) = 0
pour tout
0≤q≤qc, τχ(q) = τµ(q) + t
pour tout
q≤0, τχ(q)≥τµ(q) + t.
(2.19)
En particulier, pour tout
h≥0,
on a
τ∗
χ(h)≤


hqc
si
0≤h≤λc+t
τ∗
µ(h−t)
si
h≥λc+t.
(2.20)
2. Le sepectre multifractal
Dχ
de
χ
satisfait



pour tout
0< h ≤λc+t, Dχ(h) = hqc
pour tout
λc+t < h ≤τ′
µ(0+) + t, Dχ(h) = τ∗
µ(h−t).
(2.21)
3. Pour tout
h > τ′
µ(0+) + t,
on a
dimH(‹
Eh−t
µ∩ {x: ∆x= 1})≤Dχ(h)≤τ∗
µ(h−t).
Remarque 2.4.4.

Des exemples de mesures satisfaisant les hypothèses du Théorème 2.4.3 sont
donnés par le Corollaire 4.3.27.

20 CHAPTER 2: INTRODUCTION EN FRANÇAIS

τ′
µ(qc)
existe si
S
vérie AWSC et n'a pas d'overlaps exacts, voir Théorème
4.3.39.

Supposons en plus des hypothèses du Théorème 2.4.3 que pour
h≥τ′
µ(0+),
il
existe
mh∈ M(Rd)
telle que
dimH(mh) = τ∗
µ(h)
et
mh(‹
Eh
µ)>0,
alors
Dχ(h+t) = τ∗
µ(h)
et
χ
satisfait le formalisme multifractal en
h+t
.
Remarquons que l'analyse multifractale est en toute généralité ardue pour
h
grand. Ce phénomène n'est pas étonnant puisque l'on peut construire des
exemples de mesures auto-similaires associées à des systèmes satisfaisant des
hypothèses de séparations très raisonnables mais qui ne satisfont pas le formal-
isme multifractal pour
h
grand (voir [72]).
2.5 Conclusion et perspectives
Comme expliqué dans cette introduction, les principes de transferts de masse
(où les théorèmes d'ubiquité) sont des outils servant en théorie métrique des
nombres, en systèmes dynamiques et en analyse multifractale. Ainsi, un grand
nombre de questions et d'extensions possibles se dégagent naturellement de ces
travaux:

Que peut-on dire du contenu
µ
-essentiel d'autres mesures que les mesures
associées à des IFS faiblements conformes? Que dire lorsque l'IFS est
par exemple auto-ane? Que peut-on dire sous la seule hypothèse que la
mesure est exactement dimensionnelle et ergodique?

Que peut-on dire de la dimension des shrinking targets associées à ces
systèmes dynamiques?

Que dire de la mesure
χ
satisfaisant (2.16) lorsque l'IFS sous-jacent est
auto-ane, satisfait l'hypothèse de séparation exponentielle ou bien est
simplement donné par une orbite d'un système dynamique muni d'une
mesure ergodique?

Quels ensembles fractals peut-on distinguer ou caractériser en consider-
ant l'approximation par des rectangles ou des ensembles d'autres formes

2.5 CONCLUSION ET PERSPECTIVES 21
au lieux de boules? De telles questions apparaissent naturellement par
exemple lorsqu'on regarde l'approximation diophantienne multiplicative.
Plus précisément, étant donné
δ≥1,
que peut-on dire de la dimension,
de la mesure de Hausdor (éventuellement pour une jauge adaptée) de
l'ensemble
Aδ=((x1, ...., xd) : ∃(p1, ..., pd)∈Nd,
d
Y
i=1 |xi−pi
q| ≤ q−δ, q ∈N
i.s.
),
ou i.s. signie que l'égalité ci-dessus à lieu pour une innité d'entiers
q.
Notons que dans ce cas-ci, les ensembles approximants sont des hyper-
boles.
Nous commencerons dans le Chapitre 4par rappeler et préciser les no-
tions requises pour comprendre ce manuscrit. Ensuite, au Chapitre 5, nous
donnerons une première construction qui est instructive concernant le résultat
d'ubiquité que l'on souhaite montrer.
Au Chapitre 6, nous dénirons la quantité géométrique au centre de notre
théorème d'ubiquité général. Etant données une suite de boules
(Bn)n∈N
et une
mesure
µ∈ M(Rd),
nous établirons aussi un lien techniquement utile entre véri-
er
µ(lim supn→+∞Bn)=1
et vérier une certaine propriété de recouvrement
pour
(Bn)n∈N
. Lorsque
(Bn)n∈N
vérie cette propriété, nous dirons que
(Bn)n∈N
est
µ
-asymptotiquement couvrante.
Dans le Chapitre 7, nous démontrons notre résultat principal d'ubiquité,
le Théorème 2.1.6 et nous l'appliquons aux mesures faiblements conformes.
Dans le chapitre 8, nous étudions l'optimalité des théorèmes d'ubiquités
établis. En particulier, nous montrons que les Corollaires 2.1.8 et 2.1.9 sont,
en un certain sens, optimaux.
Les Chapitres 9,10 et 11 donnent respectivement des applications en
théorie métrique des nombres, en systèmes dynamiques ainsi qu'en analyse
multifractale. On y montre les Théorèmes 2.3.1, Théorème 2.2.3 et Théorème
2.4.3.

22 CHAPTER 2: INTRODUCTION EN FRANÇAIS

Chapter 3
Introduction
The theory of metric approximation aims at describing geometrically (often by
computing Hausdor dimensions) the set of elements approximable by a family
of points
(xn)n∈N
chosen in advance in a metric space
(X, d)
. Historically, this
theory was born from the work of Dirichlet about the approximation of real
numbers by rational numbers in the XIXth century. This work was completed
and extended around 1920, by Jarnik and Besicovitch [47] and recently, around
2000, by the work of Jaard [46], Beresnevich and Velani [11]. It is only since
1990 that certain problems in Diophantine approximation were formulated in
terms of approximation by orbits of dynamical systems [42]. In those settings,
one studies the case where the family of points
(xn)n∈N
is an orbit, i.e,
xn=
Tn(x)
,
x∈X
, where
X
is a metric space and
T:X→X
is a measurable
mapping. For examples where
T
is the doubling map on the torus, a Markovian
expanding map on the torus or other examples of expanding maps, see [33,55,
65].
Those sets also plays an important role in multifractal analysis. For exam-
ple, the study of Riemann's series [46] depends on the speed of approximation
of a real number by rational numbers. For other work mixing multifractal anal-
ysis and Diophantine approximation, let us refer to the work of Barral, Seuret,
Jaard, Liao, Fan, Shmeling Troubetzkoy and Persson [10,33,45,55,64].
3.1 Ubiquity theorems
3.1.1 An historic of ubiquity theorems
Fix
d∈N∗
and endow
Rd
with the norm
||(x1, ..., xd)||∞= max1≤i≤d|xi|
(in
this manuscript the choice of the norm actually matters very little).
One way to give meaning to the notion of speed of approximation of a

24 CHAPTER 3: INTRODUCTION
point
x
by a sequence of points
(xn)n∈N
of
Rd
, and to characterize this speed, is
to provide sequences of sets
(Un(δ))n∈N,
where
δ≥1
(sometimes one can take
a multi-parameter
δ∈[1,+∞)k
, where
k∈N
is xed) verifying:

for every
n∈N,
for every
δ≥1, xn∈Un(δ),

for every
δ≥1,|Un(δ)| → 0,

for every
δ′> δ,
for every
n∈N, Un(δ′)⊂Un(δ).
For each
δ
, we are interested in the sets
Eδ
of points
x
for which there exists
an innity of integers
(nk)k∈N
such that
x∈Unk(δ).
The set of all those points
is called the limsup
1
of the sequence
(Un(δ))n∈N
and can be written as
Eδ= lim sup
n→+∞
Un(δ) := \
N∈N[
n≥N
Un(δ).
We say that a point
x
is approximable at speed
δ
if
x∈Fδ:= Eδ\Sδ′>δ Eδ′
and we aim in general at computing the Hausdor dimensions (see Section
4.2, Lemma 4.2.3 for a denition of the Hausdor dimension)
dimH(Eδ)
and
dimH(Fδ)
(in practice, one can derive the value of
dimH(Fδ)
when one knows
Hζ(Eδ)>0
where
ζ
is an appropriate gauge and
Hζ
is the Hausdor measure
associated with
ζ
).
The rst historical example of such computations is the theorem of Jarnik-
Besicovitch.
Theorem 3.1.1
(Jarnik-Besicovitch [47])
.
For
x∈R
and
r > 0,
denote by
B(x, r)
the closed ball centered in
x
and of radius
r.
Then, for every
δ≥1,
denoting
Eδ:= lim supq∈N,0≤p≤qB(p
q,1
q2δ)
and
Fδ:= Eδ\Sδ′>δ Eδ′
, one has:
dimH(Eδ) = dimH(Fδ) = 1
δ.
(3.1)
The reader interested in other works around rational approximation may
refer to [11,15,16,27] (homogeneous and inhomogeneous rational approxima-
tion) and [1,17,18,54] (rational approximation of elements of the middle-third
Cantor set).
Usually, to compute the Hausdor dimension of those sets, one proceeds
separately for the lower-bound and the upper-bound. In practice, getting an
1
Remark here that the denition of
lim supn→+∞Un
does not depend on the manner we
enumerate the sets
{Un}n∈N
in the sens that for any bijective mapping
ϕ:N→N,
one has
lim supn→+∞Un= lim supn→+∞Uϕ(n).
In particular later on in the manuscript, there might
be some limsup sets taken over families of sets for which the enumeration we choose is not
explicitly stated.

3.1 UBIQUITY THEOREMS 25
upper-bound for
dimH(Eδ)
is often either straightforward, either very complex
(the reader will notice that in this manuscript, the upper-bound for
dimH(Eδ)
are often much easier to establish than the corresponding lower-bound).
The ubiquity theorems, also called mass transference principles ([11]),
are tools built to provide lower-bounds for
dimH(Eδ).
Those theorems work in the following way: one look for a measure
µ∈
M(Rd) := 
probability measures on
Rd
(hopefully adapted to our problem)
such that
µ(E1) = 1.
Then, using the geometric property of the measure
µ
(self-
similarity, multifractal analysis for example), one establishes a lower-bound for
dimHEδ
for every
δ > 1.
Here is an important example of such theorems. One
says that a measure
µ∈ M(Rd)
is
s
-Alfhors regular, where
0≤s≤d,
if
there exists two constants
0< C1≤C2
such that, for every ball
B(x, r)
with
x∈supp(µ) :=
the topological support of
µ
and
0< r ≤1,
we have
C1rs≤µ(B(x, r)) ≤C2rs.
The Lebesgue measure
Ld
on
[0,1]d
is an example of such a measure.
Theorem 3.1.2
(Beresnevich-Velani [11])
.
Let
0≤s≤d
be a real number and
µ∈ M(Rd)
an
s
-Alfhors-regular measure. Let
(Bn)n∈N
be a sequence of closed
ball centered on
supp(µ)
and satisfying:

|Bn| → 0,

µ(lim supn→+∞Bn)=1.
Let
δ > 1
be a real number and
B
a ball centered on
supp(µ).
For
s0≥0
,
denote by
Hs0
the Hausdor measure of dimension
s0
(see
(4.2.2)
) and for
n∈N
,
Bδ
n=B(xn, rδ
n)
if
Bn=B(xn, rn).
One has
Hs
δ(lim sup
n→+∞
Bδ
n∩B)=+∞.
In particular, we recover a result previously established by Jaard ([46] 1999),
dimH(lim sup
n→+∞
Bδ
n)≥s
δ.
In the case
µ=Ld
, one can also deal with the more general situation
where instead of giving a lower-bound for the dimension of
lim supn→+∞Bδ
n
for
every
δ > 1,
one describes a geometric quantity, the Hausdor content (see
Section 4.2, equation (4.7)), to estimate in order to get accurate lower-bounds
for
dimH(lim supn→+∞Un)
where
(Un)n∈N
is any sequence of open sets satisfying
Un⊂Bn.

26 CHAPTER 3: INTRODUCTION
Theorem 3.1.3
(Koivusalo-Rams [51])
.
Let
(Bn)n→+∞
be a sequence of balls
of
[0,1]d
verifying
|Bn| → 0
,
(Un)n∈N
a sequence of opens sets satisfying for
every
n∈N, Un⊂Bn
and
0≤s≤d
a real number.
Assume that:

for every
n∈N
large enough,
Hs
∞(Un)≥ Ld(Bn),

Ld(lim supn→+∞Bn) = 1.
Then , one has
dimH(lim sup
n→+∞
Un)≥s.
One should mention here that previous works on limsup generated by ran-
dom sets already highlighted the role of the Hausdor content to characterize
the dimension of those limsup sets [28,38,48,49] and recently, some interesting
techniques based on the Hausdor where also developed in [29].
One can for instance take
Un
an open rectangle for every
n∈N
. More
precisely, x
(τ1, ..., τd)∈[1,+∞)d
and let
(Bn:= xn+Qd
i=1[−rn, rn])n∈N
be a
sequence of closed balls of
[0,1]d
verifying
|Bn| → 0
and
Ld(lim supn→+∞Bn) =
1.
Set
Rn=xn+
d
Y
i=1
(−rτi
n, rτi
n).
(3.2)
Then, for every
0≤s≤d,
the
s
-dimensional Hausdor content of the the
rectangles
Rn
can be explicitly determined and depends on
rn
and the numbers
τ1, ..., τd
up to some multiplicative constant
C
which only depends on
d
. As a
consequence of this computation, one obtains that, for
s0= min1≤i≤d
d+P1≤i≤dτk−τi
τk
,
Hs0
∞(Rn)≥Crd
n≥CLd(Bn).
It is also easily veried that this implies that for every
s<s0
and every large
enough
n
,
Hs
∞(Rn)≥rd
n=Ld(Bn),
Applying Theorem 3.1.3, one obtains
dimH(lim sup
n→+∞
Rn)≥s0= min
1≤i≤d
d+P1≤i≤dτk−τi
τk
.
When the measure is not homogeneous, the situation is more complex.
Barral-Seuret Theorem deals with the case where the measure is self-
similar satisfying the Open Set Condition (OSC in short, see Section 4.3, Def-
inition 4.3.3 for more details about the dierent separation conditions). For
µ

3.1 UBIQUITY THEOREMS 27
a self-similar measure, denote by
dim(µ)
the dimension of
µ
(see Section 4.2,
Denition 4.2.12 and Section 4.3, Theorem 4.3.15).
Theorem 3.1.4
(Barral-Seuret [8])
.
Let
µ∈Rd
be a self-similar satisfying the
OSC and
(Bn)n∈N
a sequence of balls centered on
supp(µ)
verifying
|Bn| → 0.
If
µ(lim sup
n→+∞
Bn)=1,
then, for every
δ≥1,
dimH(lim sup
n→+∞
) = dim(µ)
δ.
(3.3)
In this manuscript, we develop a tool which plays the same role as the
Hausdor content in Theorem 3.1.3 but in the case where
µ
is any probability
measure of
M(Rd)
. We will show in particular that the resulting ubiquity
theorem extends both the Theorems 3.1.3 and 3.1.4.
3.1.2 A general ubiquity theorem
Let us start by introducing the following geometric quantity.
Denition 3.1.5
(D. [25])
.
Let
µ∈ M(Rd)
be a probability measure,
0≤s≤d
and
A⊂Rd
a measurable set. The
s
-dimensional
µ
-essential Hausdor content
of
A
is dened as
Hµ,s
∞(A) = inf {Hs
∞(E) : E⊂A, µ(E) = µ(A)}.
(3.4)
For
µ∈ M(Rd)
, denote by
dimH(µ)
the lower Hausdor dimension of
µ
(dened in Section 4.2, Denition 4.2.12). The ubiquity theorem associated
with the essential Hausdor content is the following (the version stated here is
simplied compared to Theorem 7.1.2).
Theorem 3.1.6
(D. [25])
.
Let
µ∈ M(Rd)
be a probability measure,
(Bn)n∈N
a
sequence of balls
|Bn| → 0
,
(Un)n∈N
a sequence of open sets satisfying for every
n∈N
,
Un⊂Bn
and
0≤s≤dimH(µ).
Assume that

one has
µ(lim sup
n→+∞
1
2Bn) = 1,

for every
n
large enough,
Hµ,s
∞(Un)≥µ(Bn).

28 CHAPTER 3: INTRODUCTION
Then, one has
dimH(lim sup
n→+∞
Un)≥min {s, dimH(µ)}.
(3.5)
In Chapter 7, the result we show is actually slightly more general than Theorem
3.1.6.
Of course, to apply Theorem 3.1.6 to a measure
µ
, one needs to under-
stand this measure nely enough so that one can estimate precisely
Hµ,s
∞(Un)
for
every
0≤s < dimH(µ)
and each set
Un
. It turns out that when the measure
is weakly conformal (Section 4.3, Denition 4.3.1), it is possible to estimate
precisely the
s
-dimensional Hausdor content of every open set and every
s
without any separation assumption on the IFS
.
Theorem 3.1.7
(D. [23])
.
Let
µ∈ M(Rd)
be a weakly conformal measure and
K
its support.
For every
0≤s < dim(µ)
, for every
ε > 0
, there exists a constant
c=c(d, µ, s, ε)>0
which depends on
d
,
µ
,
s
and
ε
, such that for every closed
balls
B=B(x, r)
centered on
K
, for every
0< r ≤1
and for every open set
Ω
, one has
c(d, µ, s, ε)|B|s+ε≤ Hµ,s
∞(˚
B)≤ Hµ,s
∞(B)≤ |B|s
c(d, µ, s, ε)Hs+ε
∞(Ω ∩K)≤ Hµ,s
∞(Ω) ≤ Hs
∞(Ω ∩K).
(3.6)
For every
s > dim(µ)
,
Hµ,s
∞(Ω) = 0.
Combining Theorem 3.1.6 and 3.1.7 we recover Theorem 3.1.3 by taking
µ=Ld
and we obtain the following corollaries.
Corollary 3.1.8.
Let
µ∈ M(Rd)
be a weakly conformal measure and
(Bn)n∈N
a sequence of balls centered on
supp(µ)
verifying
|Bn| → 0.
If
µ(lim sup
n→+∞
Bn) = 1,
then for every
δ≥1,
dimH(lim sup
n→+∞
Bδ
n)≥dim(µ)
δ.
Corollary 3.1.9.
Let
µ∈ M(Rd)
be a weakly conformal measure with
supp(µ) =
[0,1]d
and
(Bn)n∈N
a sequence of closed ball of
[0,1]d
verifying
|Bn| → 0.
Let
(τ1, ..., τd)∈[1,+∞)d
and
Rn
, dened as in
(3.2)
.

3.1 UBIQUITY THEOREMS 29
If
µ(lim sup
n→+∞
Bn) = 1,
then
dimH(lim sup
n→+∞
Rn)≥min
1≤i≤d
dim(µ) + P1≤i≤dτk−τi
τk
.
Note that Corollary 3.1.8 extends Theorem 3.1.4.
Let us also mention that for every
1≤τ1≤... ≤τd
, every sequence of ro-
tation matrices
(Ok,n)k∈N,0≤n≤2k−1∈ Od(R)N,
almost every sequence of realiza-
tion of i.i.d. uniformly distributed random variables on
[0,1]d
,
(xk,n)k∈N,0≤n≤2k−1
,
it is proved in [48] (and the proof uses implicitly the Corollary 3.1.9 in the case
µ=Ld
) that the family
R=(xk,n +Ok,n
d
Y
i=1
(−2−kτi,2kτi),0≤n≤2k−1, k ∈N)
veries
dimH(lim sup
R∈R
R) = min
1≤k≤d(d+Pk
j=1 τk−τj
τk).
In particular, the bound obtained in Corollary 3.1.9 is optimal. We will
see later on that it is indeed optimal in a strong sens.
Corollaries 3.1.8 and 3.1.9 shows how Theorem 3.1.6 is used: to get a
lower-bound for
dimH(lim supn→+∞Un)
, one needs to establish for every
n∈N
large enough a relation which only depends on
Bn
,
Un
and
µ
.
In Chapter 8, we study the optimality of the bound provided by Theorem
3.1.6.
3.1.3 Study of optimality in ubiquity theorems
The lower-bounds established in Theorems 3.1.3,3.1.4 and 3.1.6 are meaningful
only if they are, in a sens that is satisfying enough, optimal. One way to deal
with this question is to show that, if one does not have more information (com-
pared to the hypotheses of those theorems) about the sequence
(Bn)n∈N
, there
is no reason for
dimH(lim supn→+∞Un)
to be larger than the bound provided.
We can show this by extracting from the sequence
(Bn)n∈N
, when possible, a
sub-sequence
(Bϕ(n))n∈N
which still veries the hypotheses of the ubiquity the-
orems and such that
dimH(lim supn→+∞Uϕ(n))
is precisely the bound given. In
this situation, we are ensured that one can not obtain a better bound in general
under the same hypotheses.

30 CHAPTER 3: INTRODUCTION
Let us introduce the following denition, which is equivalent to the no-
tion of weakly redundant sequence of balls introduced by Barral and Seuret
(Denition 8.1.1).
Denition 3.1.10.
Let
B= (Bn)n∈N
be a sequence of balls verifying
|Bn| → 0.
For
k∈N
, dene
Tk(B) = Bn∈ B : 2−k−1≤ |Bn|<2−k
(3.7)
Nk(B) = max
B∈Tk(B)#{B′∈ Tk(B) : B∩B′=∅}.
One says that
B
is weakly redundant if
lim
k→+∞
log Nk(B)
k= 0.
Heuristically, a sequence
(Bn)n∈N
is weakly redundant if, for each
k∈N
,
each ball
Bn
satisfying
|Bn| ≈ 2−k
intersects less than
2o(k)
other ball
Bn′
satisfying also
|Bn′| ≈ 2−k.
We emphasize that the weak redundancy property
is a natural assumption to study the optimality of the bounds provided by
ubiquity theorems but this condition is related only to the sequence of balls
itself (no measure is involved) and an important consequence of this property
is that, for every measure
µ∈ M([0,1]d)
and every
ε > 0,
we have
X
n∈N|Bn|εµ(Bn)<+∞.
(3.8)
There are many sequences of balls which satises the weak redundancy con-
dition, such as the rational balls
ÄB(p
q,1
q2)äq∈N,0≤p≤q,q∧p=1 ,
but there are also
examples of sequences of balls, natural to consider, which does not satisfy this
condition. It is for instance the case of
ÄB(p
q,1
q2)äq∈N,0≤p≤q.
However, one can
show that, in many situations, those sequences which does not satisfy the weak
redundancy property still satises (3.8) for a measure characterizing well the
system (it is the case for
ÄB(p
q,1
q2)äq∈N,0≤p≤q
with the Lebesgue measure). Let
us also mention that there are other hypotheses used by dierent authors, that
are comparable to the weak redundancy but slightly dierent. For example,
the notion of optimal regularity of sequences of an approximating sequence of
point has been used in some settings (see [15]).
In the case of a weakly conformal measure, the weak redundancy condition
allows to state an upper-bound theorem.
Theorem 3.1.11
(D.[24])
.
Let
µ∈ M(Rd)
be a weakly conformal measure,
(Bn)n∈N
a weakly redundant sequence of balls, centered on
supp(µ)
and sat-
isfying
|Bn| → 0,(Un)n∈N
a sequence of open set such that
Un⊂Bn
and

3.1 UBIQUITY THEOREMS 31
0≤s < dim(µ).
If for every large enough
n
,
Hµ,s
∞(Un)≤µ(Bn),
then
dimH(lim sup
n→+∞
Un)≤s.
Remark that, given
µ∈ M(Rd)
,
(Bn)n∈N
and
(Un)n∈N
satisfying, for every
n∈NUn⊂Bn
, it is not true in general that there exists a critical exponent
s0
such that for every
s<s0
, for every
n
large enough,
Hµ,s
∞(Un)≥µ(Bn)
and for every
s>s0,
for every
n
large enough,
Hµ,s
∞(Un)≤µ(Bn).
Nonetheless, when the measure involved is weakly conformal and the
choice of the sets
Un
is consistent (for instance, if we consider rectangles all
shrunk by the same factors
(τ1, ..., τd)∈[1,+∞)d
or balls
Bδ
n
, where
δ≥1
is
xed in advance), Theorem 3.1.11 allows to conclude that the bound provided
by Corollaries 3.1.8 and 3.1.9 are optimal.
Corollary 3.1.12.
Let
µ∈ M(Rd)
be a weakly conformal measure and
(Bn)n∈N
a sequence of balls centered on
supp(µ)
verifying
|Bn| → 0
and
µ(lim sup
n→+∞
1
2Bn) = 1.
Then
1. there exists a sub-sequence
(Bϕ(n))n∈N
verifying
µ(lim sup
n→+∞
Bϕ(n)) = 1
and for every
δ≥1,
dimH(lim sup
n→+∞
Bδ
ϕ(n)) = dim(µ)
δ.
2. if
supp(µ) = [0,1]d
, there exists a sub-sequence
(Bϕ(n))n∈N
verifying
µ(lim sup
n→+∞
Bϕ(n)) = 1

32 CHAPTER 3: INTRODUCTION
and for every
(τ1, ..., τd)∈[1,+∞)d,
dimH(lim sup
n→+∞
Rϕ(n)) = min
1≤i≤d
dim(µ) + P1≤i≤kτk−τi
τk
,
when
Rn
is dened as in
(3.2)
.
In particular, the bound given by the Corollaries 3.1.8 and 3.1.9 are opti-
mal in relatively satisfying sens: under reasonable hypotheses on the sequence
(Bn)n∈N,
one can always extract a sub-sequence
(Bϕ(n))n∈N
which still veries
µ(lim supn→+∞Bϕ(n)) = 1
and such that the limsup generated by the shrunk
sets (balls or rectangles) have the expected Hausdor dimension for every con-
traction ratios at the same time.
The next section presents some results in Diophantine approximation.
Those results are established in Chapter 9.
3.2 An application in Diophantine approxima-
tion
As mentioned at the beginning of the introduction, a classical problem in Dio-
phantine approximation consists in determining the Hausdor dimension of
various sets of points approximable by rational numbers.
Consider
f0:R→R, f1:R→R
and
f2:R→R
dened as
f0:x→1
3x
,
f1(x) = 1
3x+1
3
and
f2(x) = 2
3+1
3x
, and denote by
K1/3
the attractor (see
Section 4.3, Proposition 4.3.1) of
{f0, f2}
. A problem raised by Mahler consists
in computing the Hausdor dimension of the set
K1/3∩Fδ
(where
Fδ
is dened
as in Theorem 3.1.1) of the points of
K1/3
which are approximable by rational
numbers at speed
δ≥1
xed in advance. This problem, although intensively
studied (see [1,11,17,54] among many other works), remains unsolved to this
day but the ubiquity theorems allowed to establish some partial results.
Theorem 3.2.1
(Velani-Beresnevich [11], Barral-Seuret [8])
.
For ever
δ≥1,
dimHÇlim sup
0≤p≤q,q→+∞
B(p
q,1
q2δ)∩K1/3å≥dimH(K1/3)
2δ.
The value of
dimHÄlim sup0≤p≤q,q→+∞B(p
q,1
q2δ)∩K1/3ä,
where
δ≥1,
has been the subject of various conjectures
(1) Levesley, Salp, Velani [54]:
dimHÇlim sup
0≤p≤q,q→+∞
B(p
q,1
q2δ)∩K1/3å=dimH(K1/3)
δ.

3.2 AN APPLICATION IN DIOPHANTINE APPROXIMATION 33
(2) Bugeaud-Durand [18]:
dimHÇlim sup
0≤p≤q,q→+∞
B(p
q,1
q2δ)∩K1/3å= max ßdimH(K1/3)
δ,2
δ+ dimH(K1/3)−1™.
Thanks to Theorem 3.1.3 (which implied by Theorem 3.1.6), one can
obtain a complete characterization of points approximable by rational numbers
in a set which contains
K1/3
and has same Hausdor dimension.
Denote by
Λ = {0,1,2}
and by
σ
the usual shift on
ΛN
, dened as
σ((i1, i2, ...)) = (i2, ...)
. The canonical projection from
ΛN
on
[0,1]
is the map
π:x= (xn)n∈N7→ lim
n→+∞fx1◦... ◦fxn(0).
(3.9)
Denition 3.2.2.
Let
ϕ: ΛN→ {0,1}
be the mapping dened as



ϕ(x) = 1
si
x1= 1
ϕ(x) = 0
si
x1= 0
ou
2
and
K(0)
1/3=πÅßx∈ΛN: lim inf
k→+∞
Skϕ(x)
k= 0™ã,
where
(Skϕ)k∈N
is the Birkho average of
ϕ
dened for every
x∈ΛN
as
Sk(ϕ)(x) = 1
k
k−1
X
i=0
ϕ(σi(x)).
It is shown in this manuscript (Proposition 9.1.5) that
dimHK(0)
1/3=log 2
log 3(= dimHK1/3).
We have the following result:
Theorem 3.2.3
(D.[25])
.
For every
δ≥1,
dimHlim sup
0≤p≤q,q→+∞
B(p
q,1
q2δ)∩K(0)
1/3= min ßlog 2
log 3,1
δ™.
(3.10)
Note that there is a saturation phenomena: for every
1≤δ≤log 3
log 2
, the
dimension
dimH(lim supB∈Q Bδ∩K(0)
1/3)
is constant equal to
log 2
log 3
, then decreases
as
1
δ
when
δ≥log 3
log 2 .

34 CHAPTER 3: INTRODUCTION
In the next section, we give an application of Corollary 3.1.8 to self-similar
shrinking targets problems.
3.3 Weakly conformal shrinking targets
Let
m≥2
be an integer and
S={f1, ..., fm}
a weakly conformal IFS. Denote
by
Λ = {1, .., m}
,
Λ∗=Sk≥0Λk
and
K
the attractor of
S
. Given
x∈K,
we
study the set of elements approximable by the orbit of
x
under
S
, i.e, the sets
W(x, ψ) = lim sup
i∈Λ∗
B(fi(x), ψ(i)),
where
ψ: Λ∗→R
veries
lim|i|→+∞ψ(i) = 0
and for
i= (i1, ..., ik), fi=
fi1◦... ◦fik.
Historically, the study of such sets is related to the study of shrinking
targets on the attractor of an expanding dynamical system [42].
More precisely, assume for the sake of simplicity that
S
is self-similar and
satises for every
1≤i=j≤m,
Im
(fi)∩
Im(
fj) = ∅
. Then its attractor
K
can
be seen as the attractor associated with the expanding mapping
F
dened as
F(x) = f−1
i(x)
si
x∈
Im
(fi).
Let
δ≥1
be a real number and
ψ: Λ∗→Λ∗
the mapping dened as
ψ(i) =
cie−nδ.
The set
W(x, ψ)
can be rewrote as
W(x, ψ) = z:Fn(z)∈B(x, e−nδ )
for an innity of integer
n∈N,
(3.11)
which are precisely the sets introduced and studied by Hill and Velani in [42].
There are, a priori, several natural classes of mappings one could choose
for
ϕ
:

From a dynamical standpoint, it is natural to consider
ψ
of the form
ψ(i) = |fi(K)|δ,
where
δ≥1,
since one naturally has
lim sup
i∈Λ∗
B(fi(x),|fi(K)|) = K.
In this case, Theorem 3.1.6 can be used to prove the following result:

3.3 WEAKLY CONFORMAL SHRINKING TARGETS 35
Theorem 3.3.1
(D. [23])
.
Denote by
dim(S)
the conformality dimension
of the IFS
S
(Denition
(4.3.19)
).
Assume that que
dimH(K) = dim(S),
then, for every
δ≥1
and every
x∈K
, one has
dimHÇlim sup
i∈Λ∗
B(fi(x),|fi(K)|δ)å=dimH(K)
δ.
(3.12)

In Diophantine approximation, one often prefers to consider mappings
ψ
of the form
ψ(i) = |fi(K)|g(|i|)
where
g:N→R
is decreasing. For
studies of the sets
W(x, ψ)
under those settings and the OSC, we refer to
the work of Hill and Velani and Allen and Bárány [2,42].
This dierence of point of view is justied by (3.11) and comes as well
from the fact that one often aims at proving analogs of Khintchine's
Theorem for rational numbers in the settings of IFS. To be more explicit,
let us recall Khintchine's Theorem. Denote by
ϕ:N→N
the Euler
mapping, dened as
ϕ(q) = # {1≤p≤q:p∧q= 1}.
Theorem 3.3.2
(Khintchine [50])
.
Let
ψ:N:R+
be a mapping such
that
q→ψ(q)
q
is monotonic. then
LdÇlim sup
q∈N,0≤p≤q,p∧q=1
B(p
q,ψ(q)
q)å= 1 ⇔X
q≥1
ϕ(q)ψ(q)
q= +∞.
Let us mentioned that it was conjectured by Dun et Schaeer that Theo-
rem 3.3.2 remains valid without the monotonicity assumption of
q→ψ(q)
q
and this conjecture was solved recently by Maynard and Koukoulopoulos
[52].
To make the analogy between Khintchine's Theorem and comparable re-
sult in the settings of IFS, assume to simplify that
S
is self-similar and
call
0< c1, ..., cm<1
the contraction ratios of
f1, ..., fm.
Remember that
in the case of self-similar IFS,
dim(S)
is the real number solution to
X
1≤i≤m
cdim(S)
i= 1.
One always has a natural measure
ν
on
ΛNν
dened for every
i=

36 CHAPTER 3: INTRODUCTION
(i1, ..., ik)∈Λ∗
as
ν(i) = cdim(S)
i.
Consider
µ=ν(π−1)
, where
π
is the canonical projection of
ΛN
on
K
(see
(3.9) below). One aims at establishing dichotomies of the following form
(
s≥0
is not specied here):



if
Pi∈Λ∗ψ(i)s= +∞,
then
µ(W(x, ψ)) = 1
if
Pi∈Λ∗ψ(i)s<+∞,
then
µ(W(x, ψ)) = 0.
That in mind, it is clear that
ψ
can not be arbitrary and that
ψ(i)
must
depend on the generation of the word
i
if one hopes to cover a set of large
measure by balls
{B(fi(x), ψ(i))}i∈Λn
for an innity of integers
n∈N.
For examples of such theorems in the self-similar case, one refers to [54] and
we mention the following result:
Theorem 3.3.3
(Baker [3])
.
Assume that
S
satises one of the following con-
dition


−Pm
i=1 cdim(S)
ilog(cdim(S)
i)<−2 log(Pi≤i≤mc2 dim(S)
i)
c1=... =cm.
Let
ψ: Λ∗→R
be a mapping of the form
ψ(i) = |fi(K)|g(|i|)
where
g:N→R
is positive and decreasing. If
X
n≥0
ng(n)dim(S)= +∞,
then for every
x∈K,
µ(W(x, ψ)) = 1.
Theorem 3.3.3 combined with Theorem 3.1.6 allows to prove the following
result:
Theorem 3.3.4
(D. [23])
.
Let
g:N→(0,+∞)
be a positive and decreasing
mapping and set
sg= inf 

s≥0 : X
k≥0X
i∈Λk
k(|fi(K))|g(k))s<+∞

.
(3.13)
Then, under the hypotheses of Theorem 3.3.3, if furthermore
dim(µ) = dim(S),
for every
δ≥1,
one has
dimHlim sup
i∈Λ∗
B(fi(x),(|fi(K)|g(|i|))
δsg
dim(S))=dim(S)
δ.
(3.14)

3.4 APPLICATION TO DISCRETE MEASURE 37
Remark that (3.14) provides a complete characterization of the dimension
of the sets
W(x, ψ)
for
ψ(i) = (|fi(K)|g(|i|))s
, for every
s≥0
and every
g:N→N
positive and decreasing.
The next section presents an application in multifractal analysis of dis-
crete measures, developed in Chapter 11.
3.4 Multifractal analysis of discrete measures
Let
m≥2
be an integer and
S={f1, ..., fm}
of
m≥2
contracting similarities
of contraction ratios
0< c1, ..., cm<1
and
K
the attractor of
S
. Given
(q1, ...qm+1)∈(0,1)m+1
a probability vector, we study the probability measure
solution to the following equation
χ(·) = X
1≤i≤m
qiχ(f−1
i(·)) + qm+1δx0,
(3.15)
where
δx
denotes the dirac measure at
x.
Such measures were introduced by
Olsen and Snigireva in [70] and are called in-homogeneous self-similar measures
with condensation measure
δx0
.
We can show (see Proposition 11.1.1) that such measures are discrete can
be written as
χ(·) = CX
i∈Λ∗
pict
iδfi(x0),
(3.16)
where
t > 0,(p1, ..., pm)
is an other probability vector,
C > 0
is a renormaliza-
tion constant so that
χ
is a probability measure.
Its worth mentioning that comparable measures where obtained and stud-
ied by Mandelbrot and Riedi in [56] and by Barral and Seuret in [9] by inversing
the distribution function associated with Gibbs measure or self-similar measures
satisfying the open set condition. The multifractal properties of those measures
have been studied by Falconer [31], Barral and Seuret [7,9] and Mandelbrot
and Riedi [56] but in the last case, by considering only local dimensions that
are exact limits (not liminf), the authors missed an interesting phenomenon
highlighted by Falconer, Barral and Seuret.
The ne multifractal analysis of measures was born 1970's with the work
of Mandelbrot [57,58] and Olsen [6062]. Those studies consist, given
µ∈
M(Rd),
in computing the spectrum of the dimension of the following sets (

38 CHAPTER 3: INTRODUCTION
Section 4.2, Denition 4.2.12)
Eh
µ={x∈supp(µ) : dim(µ, x) = h},
‹
Eh
µ=x∈supp(µ) : dim(µ, x) = dim(µ, x) = h,
(3.17)
i.e, computing for every
h∈R,



Dµ(h) = dimH(Eh
µ)
and
‹
Dµ(h) = dimH(‹
Eh
µ).
One generally aims at comparing the dimensions of those sets to the
Legendre transform of a scaling function, denoted by
τµ
((4.19)) and dened
by, for all
q∈R
,
τµ(q) = lim inf
n→+∞
−log PD∈Dn:µ(D)=0 µ(D)q
n
where, for
n∈N,Dn
denotes the set of dyadic cubes of generation
n
.
Let
µ
be the self-similar measure associated with
S
and
(p1, ..., pm)
, i.e.
the measure solution to the equation
µ(·) =
m
X
i=1
piµ(f−1
i(·)).
We will see that the multifractal analysis of the measure
χ
, given by
(3.16), depends both on the multifractal analysis of the self-similar measure
µ
and on the speed of approximation of a point
x
by the orbit
(fi(x0))i∈Λ∗.
Let us
introduce the degree of approximation of a point with respect to
(fi(x0))i∈Λ∗
.
Denition 3.4.1.
The degree of approximation of a point
x∈K
with respect
to
(fi(x0))i∈Λ∗
, denoted
∆x
, is dened as
∆x= sup ß∆ : x∈lim sup
n→+∞
B(fi(x0),(2ci)∆)™.
In the case where the base point of (3.16)
x0
does not belong to the
attarctor
K
of
S
, every points of
supp(χ)
outside the set
{fi(x0)}i∈Λ∗
( which
has Hausdor dimension equal to
0
) are badly approximated by those points,
i.e, have degree equal to
1
. In that case, the multifractal spectrum of
χ
is a
translated of the multifractal spectrum of
µ.
Theorem 3.4.2
(D.[26])
.
Consider
x0/∈K.
Then, for every
t > 0
, the measure

3.4 APPLICATION TO DISCRETE MEASURE 39
χ
dened as
(3.16)
exists. Moreover, for every
h≥0,
Dχ(h) = Dµ(h−t).
(3.18)
Remark that this result holds without any separation hypotheses on the
system
S
. A direct consequence of this result is that, when the attractor
K
has Lebesgue measure equal to
0
, for Lebesgue almost every
x0,
the spectrum
of the measure
χ
is a translation by
t
of the spectrum of
µ
. In other word, to
simulate a measure which has the same spectrum has
µ
but translated by
t
, it
is enough to pick randomly
x0
according to the uniform law and to simulate
the measure
χ
associated (which is discrete). Let us add that is established in
this manuscript that in this case,
χ
does not satisfy the multifractal formalism
(one actually has
τχ(q) = 0
for
q≥1
).
In the case
x0∈K
, Corollary 3.1.8 allows, under certain hypotheses, to
compute the spectrum of the measure
χ
.
Theorem 3.4.3
(D. [26])
.
Let
m≥2
,
S={f1, ..., fm}
be a self-similar IFS
satisfying AWSC (Denition 4.3.3) with no exact overlaps and
K⊂Rd
the
attractor of
S
. Let
(p1, ..., pm)
be a probability vector,
µ
the self-similar measure
associated with
(p1, ..., pm), x0∈K
and
C > 0
the normalizing constant such
that the measure
χ(·) = CX
i∈Λ∗
ct
ipiδfi(x0),
is a probability measure.
Dene
qc= min {q:τµ(q) + qt = 0}
and
λc=τ′
µ(qc)
. Then, recalling
(4.16)
:
1. The mapping
τχ
satises









for every
q≥qc, τχ(q)=0
for every
0≤q≤qc, τχ(q) = τµ(q) + t
for every
q≤0, τχ(q)≥τµ(q) + t.
(3.19)
In particular for every
h≥0,
one has one has
τ∗
χ(h)≤


hqc
if
0≤h≤λc+t
τ∗
µ(h−t)
if
h≥λc+t.
(3.20)

40 CHAPTER 3: INTRODUCTION
2. The multifractal spectrum
Dχ
of
χ
satises



for every
0< h ≤λc+t, Dχ(h) = hqc
for every
λc+t<h≤τ′
µ(0+) + t, Dχ(h) = τ∗
µ(h−t).
(3.21)
3. For every
h > τ′
µ(0+) + t,
one has
dimH(‹
Eh−t
µ∩ {x: ∆x= 1})≤Dχ(h)≤τ∗
µ(h−t).
Remark 3.4.4.

Some examples of self-similar measures satisfying the hypotheses of Theorem
3.4.3 are given in Corollary 4.3.27.

τ′
µ(qc)
exists when
S
satises the AWSC with no exact overlaps, see Theorem
4.3.39.

Assume in addition to the hypotheses of Theorem 3.4.3 that for
h≥τ′
µ(0+),
there exists
mh∈ M(Rd)
such that
dimH(mh) = τ∗
µ(h)
and
mh(‹
Eh
µ)>0,
then
Dχ(h+t) = τ∗
µ(h),
and
χ
satises the multifractal formalism at
h+t
.
One mentions also that the multifractal analysis of
χ
for
h
large is hard in
general. This phenomena is not surprising since there exists some examples of
self-similar measures satisfying very reasonable separation condition but does
not satisfy the multifractal formalism in the decreasing part of the spectrum
(see [72]).
3.5 Conclusion and perspectives
As explained in this introduction, the mass transference principles (or ubiq-
uity theorems) are tools used in metric number theory, dynamical systems and
multifractal analysis. Hence, there are many natural question and extensions
related to these three elds raised by this manuscript:

Can we estimate the
µ
-essential contents for a wider class of measures?
Can something general be said when the IFS is self-ane? Although
it seems likely that there exists exact dimensional ergodic measures for
which Corollary 3.1.8 should not hold, is it possible to dene a quantity

3.5 CONCLUSION AND PERSPECTIVES 41
one can estimate to compute the dimension of shrinking targets under
those very natural settings?

Can we compute the multifractal spectrum of the measure
χ
satisfying
(3.16) when the underlying IFS is self-ane, satises the exponential sep-
aration condition or is simply given by the orbit of an ergodic dynamical
system.

What fractal sets can we distinguish or characterize by considering the
approximation by rectangles or sets of other shapes? Such problems nat-
urally appears in multiplicative Diophantine approximation. More pre-
cisely, given
δ≥1,
what is the dimension and the Hausdor measure
(associated with an appropriate gauge function) of the set
Aδ=((x1, ...., xd) : ∃(p1, ..., pd)∈Nd,
d
Y
i=1 |xi−pi
q| ≤ q−δ, q ∈N
i.o.
),
where i.o. means that the inequality holds for innitely many
q
. In this
case, the approximating sets are hyperboles.
In Chapter 4, some recall about the basic notion frequently used in this
manuscript are given. It is also proved in this chapter that
C1
weakly conformal
IFS's satisfying AWSC without exact overlaps are dimension regular (Denition
4.3.20) and the scaling function of weakly conformal measures (under these
hypothesis) is computed for
q > 0
. A particular case of the general ubiquity
Theorem, Theorem 7.1.2 is treated in Chapter 5. More precisely, Chapter 5
deals with the case where the measure involved is quasi-Bernoulli on
Rd
.
The Chapter 6deals with the denition of the basic notion needed to
prove Theorem 7.1.2. In particular, given a measure
µ∈ M(Rd)
one denes
the notion of
µ
-asymptotically covering sequences of balls and the
µ
-essential
Hausdor content of a set. The essential content of open sets are also computed
when the measure involved is weakly conformal. Finally, as an application, one
derive from the Borel-Cantelli Lemma established in [12] for doubling measures
a version which is valid for every probability measure on
Rd
. Chapter 7is ded-
icated to the proof of Theorem 7.1.2 and its consequences when the measure
is weakly-conformal. The optimality of the lower-bounds established in Chap-
ter 7are discussed in Chapter 8. Chapters 9,10 and 11 gives application of
Theorem 7.1.2 respectively, in Diophantine approximation, to weakly confor-
mal shrinking targets and to the multifractal analysis of discrete self-similar
measures.

42 CHAPTER 3: INTRODUCTION
Some notations used in this article are summarized here.

3.5 CONCLUSION AND PERSPECTIVES 43
Notations Denition and references.
N
set of integers,
{0,1,2, ...}
.
Q
set of rational numbers,
¶p
q:p∈Z, q ∈N∗©
.
R
set of real numbers.
d∈N
dimension of the ambient
Rd
.
||x||∞, x = (x1, ..., xd)∈Rd
norm on
Rd
used in this manuscript and
dened by
||x||∞= max1≤i≤d|xi|
.
Dp, p ∈N
set of dyadic cubes of generation
p
of
Rd
.
See (4.17).
E
,
E⊂Rd
topological closure of
E
.
˚
E, E ⊂Rd
interior of
E
.
∂E, E ⊂Rd
boundary of
E
, dened by
∂E =E\˚
E
.
B(x, r), x ∈Rd, r > 0
closed ball with center
x
and radius
r
.
|E|, E ⊂Rd
diameter of the set
E
.
tB, B
a ball of radius
r > 0
and
t≥0
ball with same center as
B
and radius
tr
.
Bδ, B
ball of radius
r > 0
,
δ∈R
ball of same center as
B
and radius
rδ
.
L(Rk,Rp), k, p ∈N
space of linear maps from
Rk
to
Rp
.
||ℓ||, ℓ ∈ L(Rk,Rp), k, p ∈N
norm subordinated to
|| · ||∞
on both
Rk
and
Rp
of
ℓ
. See (4.21).
qℓq, ℓ ∈ L(Rk,Rp), k, p ∈N
See (4.21) as well.
Md(R)
space of
d×d
real matrices.
GLd(R)
space of
d×d
invertible real matrices.
Od(R)
space of
d×d
orthogonal real matrices.
S+
d(R)
space of
d×d
non neagative symmetric real
matrices.
S++
d(R)
space of
d×d
positive symmetric real
matrices.
f′(x), f :Rk→Rp, k, p ∈N
and
x∈Rk
dierential of
f
at
x.
Cn(Rk,Rp), n ∈N∪ ∞, k, p ∈N
space of mappings from
Rk→Rpn
times
dierentiable with continuous
nth
dierential at any
x∈Rk
.
Cn+ε(Rk,Rp), n ∈N∪ ∞, k, p ∈N
,
0<ε<1
space of mappings from
Rk→Rpn
times
dierentiable with
ε−
Hölder continuous
nth
dierential at any
x∈Rk
.
B(X)
Borel
σ
-algebra of
X
.
M(X)
space of Borel probability measure on
X
.
Ld
Lebesgue measure on
Rd
.
µ, ν, η
letters used for measures.
µ
-a.c. sequence of balls
µ
-asymptotically covering sequence of balls
. See Denition 6.1.1.
supp(µ), µ ∈ M(X)
topological support of
µ
.
infessµ, µ ∈ M(X)
essential inmum with respect to
µ
.
supessµ, µ ∈ M(X)
essential supremum with respect to
µ
.

44 CHAPTER 3: INTRODUCTION
τµ(q), µ ∈ M(Rd), q ∈RLq
spectrum of
µ
, see (4.19).
Eh
µ,‹
Eh
µ, µ ∈ M(Rd), h ≥0
level sets used to compute the
multifractal spectrum of
µ,
see
(4.15).
Hs(E), s ≥0, E ⊂Rds
-dimensional Hausdor measure
of
E
, see (4.2.2).
Ps(E), s ≥0, E ⊂Rds
-dimensional packing measure
of
E
, see (4.12).
dimH(E),dimP(E), E ⊂Rd
Hausdor and packing dimension
of
E
, see Lemma 4.2.3 and
Proposition 4.2.10.
Ld
Lebesgue measure on
Rd
.
Hs
∞(E), s ≥0, E ⊂Rds
-dimensional Hausdor content
of
E
, see (4.7).
Hµ,s
∞(E), s ≥0, µ ∈ M(Rd), E ⊂Rd
s
-dimensional
µ
-essential
Hausdor content of
E
, see
Denition 6.2.1.
dim(µ, x),dim(µ, x), µ ∈ M(Rd), x ∈supp(µ)
inferior and superior local
dimension of
µ
at
x
, see
Denition 4.14.
dimH(µ),dimH(µ),dimP(µ),dimP(µ),dim(µ), µ ∈ M(Rd)
see Denition 4.14.
fi,
where
i= (i1, ..., ik)∈ {1, ..., m}k, k ∈Nfi1◦... ◦fik,
see Section 4.3.
ci, pi
where
i= (i1, ..., ik)∈ {1, ..., m}k, k ∈Nci1×... ×cik
and
pi1×... ×pik.
See Section 4.3.
ci(x)
where
i= (i1, ..., ik)∈ {1, ..., m}k, k ∈N, x ∈Rd||f′
i(x)||,
S see Section 4.3.2.1.
σ
denote the usual shift operation
on
ΛN
, see beginning of Section
4.3.
OSC, SOSC and AWSC
respectively Open Set Condition,
Strong Open Set Condition and
Asymptotically Weak Separation
Condition, See Denition 4.3.3.
P(s), s ∈R
pressure associated with a
weakly conformal IFS
S
, See
Proposition 4.3.16.

Chapter 4
Covering Lemmas, Hausdor
measures and iterated function
systems
4.1 Covering lemmas
This section is dedicated to the statement of various classical and less classical
covering lemmas used throughout this manuscript. Let us start the well-known
5r-covering lemma (see [32] Lemma 4.8 p.91).
Lemma 4.1.1
(5r-lemma)
.
Let
(X, d)
be a metric space and
(Bn)n∈N
a sequence
of balls of
X
satisfying
supn∈N|Bn|<+∞.
There exists a sub-sequence of balls
(Bϕ(n))n∈N
such that for any
n < n′, Bϕ(n)∩Bϕ(n′)=∅
and
[
n∈N
Bn⊂[
n∈N
5Bϕ(n).
(4.1)
This lemma is useful when dealing with doubling measures. However in
this manuscript, one mainly deals with measures that are not doubling. The
rest of the section is dedicated to the establishment of a modied version of the
traditional Besicovitch covering theorem more adapted for our purposes.
Proposition 4.1.2
(D. [24])
.
For any
0< v ≤1
, there exists
Qd,v ∈N⋆
, a
constant depending only on the dimension
d
and
v
, such that for every bounded
subset
E⊂Rd
and every covering
F=B(x, r(x)) : x∈E, r(x)>0
of
E
,
there exists
F1, ..., FQd,v
nite or countable sub-families of
F
such that:

∀1≤i≤Qd,v
,
∀L=L′∈ Fi
, one has
1
vL∩1
vL′=∅.

46 CHAPTER 4: COVERING LEMMAS, HAUSDORFF MEASURES
AND IFS

E
is covered by the families
Fi
, i.e.
E⊂[
1≤i≤Qd,v [
L∈Fi
L.
(4.2)
The case
v= 1
corresponds to the standard Besicovich's covering lemma
(see [59], Chapter 2, pp. 28-34 for instance).
A rst step towards Proposition 4.1.2 is the next lemma, which allows to
split a given family of weakly overlapping balls into a nite number of families
of disjoint balls.
Lemma 4.1.3.
Let
0< v < 1
and
B= (Bn)n∈N
be a countable family of balls
such that
limn→+∞|Bn|= 0
, and for every
n=n′∈N
,
vBn∩vB′
n=∅
.
There exists
γd,v + 1
(
γd,v
being the constant appearing in Lemma 4.1.4
below) sub-families of
B
,
(Fi)1≤i≤γd,v+1
, such that:

B=S1≤i≤γd,v+1 Fi
,

∀1≤i≤γd,v + 1
,
∀L∩L′∈ Fi
, one has
L∩L′=∅.
Proof.
The proof is based on the following lemma, whose proof can be found
in [59], Lemma 2.7, pp.30 - there, the result is obtained for
v= 1/2
but the
proof remains valid for any
v < 1
.
Lemma 4.1.4.
For any
0< v ≤1
there exists a constant
γv,d >0
depending
only on
v
and the dimension
d
only, satisfying the following: if a family of balls
B= (Bn)n∈N
and a ball
B
are such that

∀n≥1
,
|Bn| ≥ 1
2|B|,

∀n1=n2≥1
,
vBn1∩vBn2=∅,
then
B
intersects at most
γv,d
balls of
B
.
The families
F1, ..., Fγd,v+1
are built recursively.
For
k∈N
, set
G(k)=L∈ F : 2−k−1<|L| ≤ 2−k.
Notice that, since
lim
n→+∞|Bn|= 0,
each
G(k)
is empty or nite.

4.1 COVERING LEMMAS 47
Observe rst that for every
k∈N
, for every ball
B∈ G(k)
, and for every
pair of balls
B1=B2∈Sk′≤kG(k′)\ {B}
, one has
vB1∩vB2=∅
as well as
|Bi| ≥ |B|
2
for
i= 1,2
.
By Lemma 4.1.4, this implies that
B
intersects at most
γd,v
balls of
Sk′≤kG(k′)\ {B}
.
To get Lemma 4.1.3, we are going to sort the balls of
Sk′≤kG(k′)
recursively
on
k
into families
F1, ..., Fγd,v+1
of pairwise disjoint balls. At each step, a new
ball
B
will be added to one of those families of balls
Fi
and the resulting family,
FiS{B}
will be denoted again by
Fi
.
Let
k0
be the smallest integer such that
G(k0)
is non-empty. Consider an
arbitrary
L0∈ G(k0)
. By Lemma 4.1.4,
L0
intersects
n0≤γd,v
other balls of
G(k0)
, that are denoted by
L1, ..., Ln0
. The sets
Fi
are then set as follows:

∀1≤i≤n0
,
Fi={Li}
,

∀n0+ 1 ≤i≤γd,v
,
Fi=∅
,

Fγd,v+1 ={L0}.
Further, consider
e
L∈ G(k0)\S0≤i≤n0{Li}
(whenever such an
e
L
exists).
The same argument (Lemma 4.1.4) ensures that
e
L
intersects at most
γd,v
balls
of
G(k0)
.
In particular there must exists
1≤i≤γd,v + 1
such that for every
L∈ Fi
,
e
L∩L=∅
. Choosing arbitrarily one of those indices
i
, one adds
e
L
to
Fi:= e
LSFi
(we keep the same name for this new family).
The same argument remains valid for any other ball
L′′ /∈S1≤j≤γd,v+1 SL∈Fj{L}
.
Hence, proceeding recursively on all balls of
G(k0)
allows to sort the balls of
G(k0)
into
γd,v + 1
families
(Fi)1≤i≤γd,v+1
of pairwise disjoint balls.
Next, let
k1
be the smallest integer such that
k1> k0
and
G(k1)
is non
empty, take an arbitrary
L(1)
0∈ G(k1)
. It is trivial to check that the family
G(k0)∪
G(k1)
and the ball
L(1)
0
satisfy the conditions of Lemma 4.1.4. Subsequently,
L(1)
0
intersects at most
γd,v
balls of
G(k0)SG(k1)
, and there must exist an integer
1≤i0≤γd,v + 1
such that
L(1)
0∩SL∈Fi0L=∅
. As before, we add this ball
L(1)
0
to the family
Fi0
.

48 CHAPTER 4: COVERING LEMMAS, HAUSDORFF MEASURES
AND IFS
Consider
e
L∈ G(k1)
such that
e
L /∈S1≤i≤γd,v+1 Fi
(whenever such a ball
exists). The exact same argument shows
e
L
intersects at most
γd,v
balls of
G(k0)SG(k1)
. Hence there exists an integer
1≤ei≤γd,v + 1
such that
L∩[
A∈Fe
i
A=∅.
One adds
e
L
to the family
Fei
, which remains composed only of pairwise disjoint
balls.
One applies this argument to every ball of
G(k1)
, hence nally sorting the
balls of
G(k0)∪G(k1)
into
γd,v + 1
families of pairwise disjoint balls, as requested.
It is now easily seen that one can proceed recursively on
k≥k0
, ending
up with the families
F1, ..., Fγd,v+1
fullling the desired properties.
We are now ready to prove Proposition 4.1.2.
Proof.
Fix
E⊂[0,1]d
and
F=B(x, r(x)) : x∈E, r(x)>0
.
One applies Besicovich's theorem (i.e. Proposition 4.1.2 with
v= 1
) to
F=B(x, r(x)) : x∈E r(x)>0
. This provides us with a nite set of families
of balls
G1, ..., Gγd,1+1
composed of pairwise disjoint balls satisfying (4.2), i.e.
E⊂S1≤i≤Qγd,1+1 SL∈GiL.
For every
1≤i≤Qd,1
, one sets
G(v)
i=1
vL:L∈ Gi
, i.e. the sets of
balls with same centers as
Gi
but with radii multiplied by
v−1>1
. Notice that
by construction,
∀1≤i≤Qd,1
,
∀L=L′∈ G(v)
i
, one has
vL ∩vL′=∅
. Hence,
Lemma 4.1.3 yields
γd,v + 1
sub-families
(G(v)
i,j )1≤j≤γd,v+1
of
G(v)
i
such that:

∀1≤j≤γd,v + 1
,
∀L=L′∈ G(v)
i,j
, one has
L∩L′=∅,

G(v)
i=S1≤j≤γd,v+1 G(v)
i,j .
Finally, we set for every
1≤i≤Qd,1
and
1≤j≤γd,v + 1
Fi,j =¶vL :L∈ G(v)
i,j ©
and
Fi=[
1≤j≤γd,v+1 Fi,j .
These sets verify that:

∀1≤i≤Qd,1
,
∀1≤j≤γd,v + 1
,
∀L=L′∈ Fi,j
,
1
vL∩1
vL′=∅
(because
the balls of
Gi,j
are pairwise disjoint),

4.2 HAUSDORFF DIMENSION, DIMENSION OF MEASURES 49

E⊂S1≤i≤Qd,1Gi=S1≤i≤Qd,1S1≤j≤γd,v+1 Fi,j .
This proves the statement and the fact that
Qd,v =Qd,1.(γd,v + 1)
.
One also recall the following density-lemma (which holds in metric sapces
in which Besicovitch's theorem holds).
Lemma 4.1.5
([13])
.
Let
m∈ M(Rd)
,
0<c<1
and
A
be a Borel set with
m(A)>0.
For every
r > 0
, set
A(r) = {x∈A:∀˜r≤r, m(B(x, ˜r)∩A)≥c m(B(x, ˜r))}
(4.3)
Then
m [
r>0
A(r)!=m(A).
(4.4)
4.2 Hausdor dimension, dimension of measures
Basic properties of the geometric measure theory are recalled in this section.
4.2.1 Hausdor measures, dimension and content
Let us start by recalling the denition of an outer measure.
Denition 4.2.1.
Let
µ
be a non negative function dened on the subsets of
Rd.
One says that
µ
is an outer measure if it veries

µ(∅) = 0,

for any
A⊂B
,
µ(A)≤µ(B),

for any sequence of sets
(En)n∈N,
µ [
n≥0
En!≤X
n≥0
µ(En)
(with still 
≤
 even when the sets
En
are disjoint measurable sets).
Denition 4.2.2.
Let
ζ:R+7→ R+
. Suppose that
ζ
is increasing in a neigh-
borhood of
0
and
ζ(0) = 0
. The Hausdor outer measure at scale
t∈(0,+∞]
associated with
ζ
of a set
E
is dened by
Hζ
t(E) = inf (X
n∈N
ζ(|Bn|) : |Bn| ≤ t, Bn
closed ball and
E⊂[
n∈N
Bn).
(4.5)

50 CHAPTER 4: COVERING LEMMAS, HAUSDORFF MEASURES
AND IFS
The Hausdor Borel measure associated with
ζ
of a set
E
is dened by
Hζ(E) = lim
t→0+Hζ
t(E).
(4.6)
For
t∈(0,+∞]
,
s≥0
and
ζ:x7→ xs
, one simply uses the usual
notation
Hζ
t(E) = Hs
t(E)
and
Hζ(E) = Hs(E)
, and these measures are called
s
-dimensional Hausdor outer measure at scale
t∈(0,+∞]
and
s
-dimensional
Hausdor measure respectively. Thus,
Hs
t(E) = inf (X
n∈N|Bn|s:|Bn| ≤ t, Bn
closed ball and
E⊂[
n∈N
Bn).
(4.7)
The quantity
Hs
∞(E)
(obtained for
t= +∞
) is called the
s
-dimensional Haus-
dor content of the set
E
. Note that it is easily seen that for
0≤s<d
,
the
s
-dimensional Hausdor content is not a measure. For any cube
B
, one
has
Hs
∞(B) = |B|s
and, if one considers a packing by disjoints smaller cubes
B1, ..., Bk,
one has
X
1≤i≤kHs
∞(Bi) = X
1≤i≤k|Bi|s=X
1≤i≤k|Bi|ds
d
> X
1≤i≤k|Bi|d!s
d
≥(Ld(B))s
d=|B|s=Hs
∞(B) = Hs
∞([
1≤i≤k
Bi).
The following lemma denes the Hausdor dimension of a set.
Lemma 4.2.3.
Let
E⊂Rd
be a subset of
Rd
. There exists a unique
0≤
dimH(E)≤d
such that



for any
s < dimH(E),Hs(E) = +∞
for any
s > dimH(E),Hs(E) = 0.
(4.8)
The number
dimH(E)
is called the Hausdor dimension of the set
E
.
A very classical, but powerful, tool to estimate Hausdor dimension is
the mass distribution principle, which is stated in the following proposition,
together with the Billingsley lemma.
Lemma 4.2.4
([32])
.
Let
E⊂Rd
be a Borel set.

Let
ζ:R+7→ R+
, be a mapping, increasing in a neighborhood of
0
and
such that
ζ(0) = 0
. Assume that there exists a Borel measure
µ∈Rd
and
r0>0
such that for any
r≤r0
and any
x∈E
µ(B(x, r)) ≤ζ(r).

4.2 HAUSDORFF DIMENSION, DIMENSION OF MEASURES 51
Then, there exists a constant
C > 0
depending only on
d
such that
Hζ(E)≥µ(E)
C.
In particular, if
µ(E)>0
, for all
r
small enough and some
0≤s≤d
,
one has
ζ(r)≤rs
then
dimH(E)≥s.

Assume that there exist
µ∈ M(Rd)
and
0≤s≤d
such that for every
x∈E, dim(µ, x)≤s.
Then
dimH(E)≤s.
Let us also give some basic properties of the Hausdor content.
Proposition 4.2.5.
Let
E⊂Rd
be a bounded set subset of
Rd
. One has, for
any
0≤s≤d
:
1.
Hs
∞(E)≤ |E|s
,
2.
t:7→ Ht
∞(E)
is a non increasing function,
3. for any
δ≥1,Hs
δ
∞(E)≥(Hs
∞(E))1
δ,
4.
Hs
∞(E)≤ Hs(E)
and
Hs
∞(E)=0⇔ Hs(E) = 0.
Proof.
Item
(1)
and
(2)
are easily derived from (4.7). Item
(3)
is a direct
consequence of Hölder's inequalities. The rst part of item
(4)
also readily
follows from (4.7) and (4.2.2). Moreover, since
Hs
∞(E)≤ Hs(E)
, one has
Hs(E)=0⇒ Hs
∞(E)=0.
We prove the second part of item
(4).
Assume that
Hs
∞(E) = 0.
Consider
t > 0
and
0< ε1
s≤t.
By (4.7), there exists a sequence of balls
(Bn)n∈N
such
that
E⊂Sn≥0Bn
and
X
n≥0|Bn|s≤ε.
(4.9)
Notice that (4.9) implies that for any
n∈N,|Bn|s≤ε,
so that
|Bn| ≤ t.
In
particular,
Hs
t(E)≤ε.
Letting
ε→0
, one gets
Hs
t(E) = 0
and letting
t→0,Hs(E) = 0.
Item
(4)
indicates that the Hausdor measure and the Hausdor content
of a set are actually close to give the same geometric information about the
distribution of the set
E
. The major dierence between those quantities is that

52 CHAPTER 4: COVERING LEMMAS, HAUSDORFF MEASURES
AND IFS
the Hausdor measure only gives information at innitesimal scales compared
to the diameter of the set
E
, while the Hausdor content gives information
about the distribution of the set
E
taking all scales lower than the diameter of
E
into account. This fact is characterized eciently by the following version
of the well-known Frostman lemma.
Proposition 4.2.6
([20])
.
Let
s≥0
. There is a constant
κd>0
depending
only on the dimension
d
such that for any bounded set
E⊂Rd
with
Hs
∞(E)>0
,
there exists a probability measure supported by
E
, that we denote by
ms
E
, such
that
for every ball
B(x, r)
,
ms
E(B(x, r)) ≤κd
rs
Hs
∞(E).
(4.10)
4.2.2 Packing measure and packing dimension
Let us start by dening the
t
-packing of a set.
Denition 4.2.7.
Let
E
be a subset of
Rd
and
t > 0.
A collection of balls
{Bi}i∈I
is said to be a
t
-packing of
E
if, for every
i=j∈I, Bi
is centered on
E
,
|Bi| ≤ t
and
Bi∩Bj=∅.
In this manuscript, the denition a packing measure associated with a
general gauge function will not be used, so that one only denes it in the
following traditional way.
Denition 4.2.8.
Let
E
be a subset of
Rd
and
0≤s≤d.
The
s
-dimensional
packing pre-measure of
E
is dened as
Ps
0(E) = lim
t→0+sup (X
i∈I|Bi|s:{Bi}i∈I
is a t-packing of E
)
(4.11)
Let us remark that this set function is not an outer-measure for
s<d
. It
is easily veried that, for any dense countable set
E={xn}n∈N,Ps
0(E)=+∞
and for every
n∈N
,
Ps
0({xn}) = 0.
To overcome this diculty, one denes the packing measure of a set
E
as
follows.
Denition 4.2.9.
Let
E
be a subset of
Rd
and
0≤s≤d.
The
s
-dimensional
packing measure of
E
is dened as
Ps(E) = inf (X
i∈NPs
0(Ei) : E⊂[
i∈N
Ei).
(4.12)
Then [32]:

4.2 HAUSDORFF DIMENSION, DIMENSION OF MEASURES 53
(1)
Ps
is an outer measure,
(2) if
E1
and
E2
are two subsets verifying
d(E1, E2)>0,
one has
Ps(E1∪E2) = Ps(E1) + Ps(E2).
It is standard that any outer measure satisfying item
(2)
is a Borel measure.
The following proposition denes the packing dimension of a set.
Proposition 4.2.10.
Let
E
be a subset of
Rd.
There exists a unique number
0≤dimP(E)≤d
such that



for any
s < dimP(E),Ps(E)=+∞
for any
s > dimP(E),Ps(E)=0.
(4.13)
The number
dimP(E)
is called packing dimension of
E
.
Remark 4.2.11.

The denition of packing measures and Hausdor mea-
sures can be adapted to any metric space
(X, d).

A set
E
is called a
Gδ
if
E
can be written as
E=Tn∈NΩk,
where the sets
Ωk
are non empty open sets. It is proved that any
Gδ
dense set
E
has
full packing dimension. In particular, when one studies sets of the form
lim supn→+∞Un,
with
Un
is an open set, denoting
K=lim supn→+∞Un
(where
X
denotes the topological closure of
X
), one always has
dimP(lim sup
n→+∞
Un) = dimP(K).
This justies that the packing dimension is not a relevant quantity to study
the size of limsup sets and explains why we only consider the Hausdor
dimension in these cases.
4.2.3 Dimension of measures
The Packing and Hausdor dimension of a Borel measures are dened as follow.
Denition 4.2.12.
Let
µ∈ M(Rd)
. For
x∈supp(µ)
, the lower and upper
local dimensions of
µ
at
x
are dened as
dim(µ, x) = lim inf
r→0+
log(µ(B(x, r)))
log(r)
and
dim(µ, x) = lim sup
r→0+
log(µ(B(x, r)))
log(r).

54 CHAPTER 4: COVERING LEMMAS, HAUSDORFF MEASURES
AND IFS
Then, the lower and upper Hausdor dimensions of
µ
are respectively dened
by
dimH(µ) = ess infµ(dim(µ, x))
and
dimP(µ) = ess supµ(dim(µ, x)).
(4.14)
It is known (for more details see [32]) that
dimH(µ) = inf{dimH(E) : E∈ B(Rd), µ(E)>0}
dimP(µ) = inf{dimP(E) : E∈ B(Rd), µ(E)=1}.
When
dimH(µ) = dimP(µ)
, this common value is simply denoted by
dim(µ)
and
µ
is said to be
exact dimensional
. In this case, for
µ
-almost every
x∈
supp(µ),
one has
dim(µ, x) = dim(µ, x) = lim
r→0+
log(µ(B(x, r)))
log(r)= dim(µ).
4.2.4 Multifractal analysis of measures
The multifractal analysis of a measure
µ∈ M(R)
aims at associating a spec-
trum which characterizes the pointwise behaviors of
µ.
The following sets are
usually the object of such study.
Denition 4.2.13.
For every
µ∈ M(Rd)
and
h∈R+
, one sets
Eµ(h) = {x∈supp(µ) : dim(µ, x) = h},
‹
Eµ(h) = x∈supp(µ) : dim(µ, x) = dim(µ, x) = h.
(4.15)
Given
h≥0,
a point
x∈supp(µ)
belongs
Eh
µ
if for any
ε > 0,
there exists
rx>0
such that for every
0< r ≤rx, µ (B(x, r)) ≤rh−ε
and there exists a
sequence of radii
rn→0
for which
µ(B(x, rn)) ≥rh+ε
n.
The multifractal spectra
one naturally associates with
µ
are
Dµ:h7→ dimH(Eh
µ)
and
‹
Dµ:h7→ dimH(‹
Eh
µ).
(4.16)
In many cases,
Dµ
can be deduced from the study of the
Lq
scaling func-
tion
τµ(q)
dened below.
For
p∈N
,
Dp
stands for the set of closed dyadic cubes of
Rd
of generation
p
, i.e.
Dp=(d
Y
i=1
[ki2−p,(ki+ 1)2−p] : ∀1≤i≤d, ki∈N).
(4.17)
When
D∈ Dp
, write also
p=p(D)
so that, for every
D∈Sp∈NDp,

4.3 ITERATED FUNCTION SYSTEMS 55
D∈ Dp(D).
Denition 4.2.14.
Let
µ∈ M(Rd)
and
q∈R
. One denes
Θµ(q, n) = X
D∈Dn:µ(D)=0
µ(D)q.
(4.18)
The
Lq
scaling function
τµ
of
µ
is dened as
τµ(q) = lim inf
n→+∞−log2(Θµ(q, n))
n∈[−∞,+∞].
(4.19)
The measure
µ
is said to verify the multifractal formalism at
h
when
Dµ(h) =
τ∗
µ(h),
where
τ∗
µ(h)
denotes the Legendre transform of
τµ
at
h
dened as
τ∗
µ(h) = inf {hq −τµ(q) : q∈R}.
(4.20)
As said above, there are numerous examples of measures satisfying the
multifractal formalism, among which a large class of self-similar measures (see
Theorem 4.3.39), Gibbs measures, ....
The basic properties of
τµ
are summarized in the following proposition.
Proposition 4.2.15
([32,41])
.
Let
µ∈Rd.

The mapping
q7→ τµ(q)
is concave and decreasing. In particular, the
set on which
τµ
does not admit a derivative is at most countable and
τµ
admits right and left derivatives for all
q
's for which
−∞ < τµ(q)<+∞
.

One has
τµ(1) = 0
and
τ′
µ(1−)≤dimH(µ)≤dimP(µ)≤τ′
µ(1+).
4.3 Iterated function systems
Let
m≥2
be an integer. An Iterated Function System (IFS in short) is a set
S={f1, ..., fm}
of mappings
fi:X→X
, where
X⊂Rd
.
Given an open set
U⊂Rd
and
f:U→Rd
a dierentiable map, for any
x∈U
:

f′(x)
is the dierential of
f
at
x
.

for
ℓ∈ L(Rk,Rd)
, one denotes by
||ℓ|| = sup
x∈Rk=0
||ℓ(x)||
||x||
and
qℓq= inf
x∈Rk=0 ||ℓ(x)||
||x||
(4.21)

56 CHAPTER 4: COVERING LEMMAS, HAUSDORFF MEASURES
AND IFS
Let us recall the following result.
Proposition 4.3.1
(Hutchinson [44])
.
Let
m≥2
be an integer,
U⊂Rd
an
open set and
S={f1, ..., fm}
a system of
C1
maps from
U
to
U.
Assume that
S
is uniformly contracting, i.e
max
1≤i≤msup
x∈U||f′
i(x)|| <1.
Then there exists a unique non empty compact set
K
satisfying
K=[
1≤i≤m
fi(K).
Moreover, for any
(p1, ..., pm)∈(0,1)m
, there exists a unique measure
µ∈
M(Rd)
supported on
K
satisfying
µ=X
1≤i≤m
piµ(f−1
i(·)).
(4.22)
From now on, an IFS designates a uniformly contracting system
of
C1
maps.
The following notations are used throughout the manuscript:

Λ(S) = {1, ..., m}
and
Λ(S)∗=Sk≥0Λ(S)k
. When there is no ambiguity
on the system
S
involved, one simply writes
Λ(S)=Λ.

KS
denotes the attractor of
S
(or simply
K
when the context is clear).

For
i= (i1, ..., ik)∈Λk
, the cylinder
[i]
is dened by
[i] = (i1, ..., ik, x1, x2, ...):(x1, x2, ...)∈ΛN.
Moreover, if
(αn)n∈N
is a sequence of real numbers, one sets
αi=αi1×... ×αik
and
fi=fi1◦... ◦fik.
For example, given the probability vector
(p1, .., pm), pi=pi1×... ×pik.

The set
ΛN
is endowed with the topology generated by the cylinders. The
set of probability measures on the Borel sets with respect to this topology
is denoted
M(ΛN)
.

4.3 ITERATED FUNCTION SYSTEMS 57

The shift operation
σ: ΛN→ΛN
is dened for any
(i1, i2, ...)∈ΛN
by
σ((i1, i2, ...)) = (i2, i3, ...).
(4.23)

The canonical projection of
ΛN
on
K
will be denoted
πΛ
(or simply
π
when there is no ambiguity) and, xing any
x∈K,
is dened, for any
(i1, i2, ....)∈ΛN
, by
π((i1, ...)) = lim
k→+∞fi1◦... ◦fik(x).
(4.24)
It is easily veried that
π
is independent of the choice of
x.
Let us recall the following classical result.
Proposition 4.3.2
([44])
.
Let
(p1, ...pm)
be a probability vector and
µ∈ M(Rd)
be the measure solution to
(4.22)
. Dene
ν∈ M(ΛN)
by setting, for any
cylinder
[i] = [(i1, ..., ik)], ν([i]) = pi=pi1×... ×pik.
Then:
1.
µ=ν◦π−1.
2.
ν
is ergodic with respect to
σ,
i.e., for any Borel set
A⊂ΛN,
one has
σ−1(A)⊂A⇒ν(A) = 0
or
ν(A)=1.
4.3.1 IFS and separation condition
In the study of the IFS's, understanding the possible overlaps between the
images of the maps involved plays a key role. Let
m≥2
and
S={f1, ..., fm}
be an IFS from an open set
U⊂Rd
to
U
.
Let us rst introduce, for all
k∈N
,
Λ(k)=i= (i1, ..., in)∈Λ∗: 2−k−1<|fi(K)| ≤ 2−k
(4.25)
and
e
Λ(k)=i= (i1, ..., in)∈Λ∗:|fi(K)| ≤ 2−k<|f(i1,...ik−1)(K)|.
(4.26)
Denition 4.3.3.
The IFS
S
is said to satisfy:
1. the strong separation condition (SSC) when for any
1≤i=j≤m
, one
has
fi(K)∩fj(K) = ∅.

58 CHAPTER 4: COVERING LEMMAS, HAUSDORFF MEASURES
AND IFS
2. the open set condition (OSC) when there exists a non empty open set
V
such that for any
1≤i=j≤m
, one has
fi(V)∩fj(V) = ∅
and
[
1≤i≤m
fi(V)⊂V.
3. the strong open set condition (SOSC) when
S
satises the open set con-
dition and the involved open set
V
veries
V∩K=∅.
4. the asymptotically weak separation condition (AWSC) [35] when, writing
for
k∈N
,
tk(S) = max
x∈Rd#¶fi:i∈e
Λ(k)
and
fi(K)∩B(x, 2−k)=∅©,
(4.27)
one has
log tk(S)
k→0.
Let us also add here that when the IFS
S
has no exact overlaps, (i.e, for
any
i=j∈Λ∗
,
fi=fj
), one also has
tk(S) = max
x∈Rd#¶i:i∈Λ(k)
and
fi(K)∩B(x, 2−k)=∅©,
(4.28)
5. the exponential separation condition (ESC) [43] when
S
consists in ane
invertible transformations and the following separation property is veri-
ed.
More precisely, dene the metric
∆
between ane invertible transforma-
tion by using the polar decomposition on the linear part and setting for
any
(O, T , a),(O′, T ′, a′)∈ Od(R)× S++
d(R)×Rd
,
∆(OT +a, O′T′+a′) = |log(||T||)−log(||T′||)|+||O−O′|| +||a−a′||.
Then
S
veries the ESC when there exists
0<c<1
such that, for any
k∈N,
any
i=j∈Λk,
one has
∆(fi, fj)≥ck.
(4.29)
Remark 4.3.4.
It is proved that IFS consisting of similarities behave well
in terms of dimension when they satisfy the ESC and additional irreducibility
properties (see Theorem 4.3.35). It is worth noticing that this separation con-
dition is not really restrictive. For instance, if one chooses an IFS on
R
with

4.3 ITERATED FUNCTION SYSTEMS 59
no exact overlaps such that its linear parts have algebraic contracting ratio and
the translation parts have algebraic coecient, then it satises the ESC [43].
Let us also mention the following result.
Theorem 4.3.5
([63])
.
Assume that the IFS
S={f1, ..., fm}
veries that for
any
1≤i≤m
, for any
x, f′
i(x)
is a similarity and
x7→ f′
i(x)
is Hölder. Then
S
veries the OSC
⇔S
veries the SOSC.
Let
(p1, ..., pm)
be a probability vector and
µ∈ M(Rd)
be the measure
solution to (4.22). When the system satises the SSC, one recovers the intu-
itive notion of self-similarity for
µ:
Let
A
be a Borel set and
i∈Λk.
Iterating
(4.22), one has
µ(fi(A)) = X
j∈Λk
piµ(f−1
j◦fi(A)).
Since, in the case of the SSC, if
j=i, fj(K)∩fi(K) = ∅
, one has
f−1
j◦fi(A) = ∅
so that
µ(fi(A)) = piµ(A) = µ(fi(K))µ(A).
(4.30)
Let us also mention the following comparable result when
S
satises the
SOSC.
Lemma 4.3.6.
Assume that
S
satises SOSC and call
V
the involved open
set. Then for any
µ∈ M(Rd)
solution to
(4.22)
,
µ(V)=1
and, for any Borel
set
A⊂V
, any
i∈Λ∗,
one has
µ(fi(A)) = µ(fi(K∩V))µ(A).
(4.31)
Proof.
Recall that
K∩V=∅
and that
supp(µ) = K.
Since
V
is open, one has
µ(V)>0.
Denote by
ν
the measure on
ΛN
dened by, for any
[i] = [(i1, ..., ik)],
ν([i]) = pi1×... ×pik.
Let
π
be the canonical projection from
ΛN
on
K
, dened by (4.24). One has
µ=ν◦π−1
, so that
ν(π−1(V)) >0.
Moreover,
σ−1π−1(V) = [
1≤i≤m[
x∈π−1(V)
(i, x) = π−1 [
1≤i≤m
fi(V)!⊂π−1(V).
Since
ν
is
σ
-ergodic, one has
ν(π−1(V)) = 1 = µ(V).
The proof of (4.31) is
similar to the proof in the case of the SSC when one restricts ourselves to the
open set V.

60 CHAPTER 4: COVERING LEMMAS, HAUSDORFF MEASURES
AND IFS
Remark 4.3.7.
Theorem 2.1.4 was only stated for self-similar measures satis-
fying the strong separation condition and for multinomial measures. The main
ingredient of the proof is actually
(4.30)
. Since
(4.31)
holds in an open set of
full measure, one easily deduces that its proof actually holds for any self-similar
measure satisfying the SOSC, which is proved to be equivalent to satisfy the
OSC (Theorem 4.3.5).
4.3.2 Generalities on conformal and weakly conformal IFS
In this section one restricts ourselves to weakly conformal IFS's. This subclass
contains in particular conformal (when the
fi
's are
C1
conformal maps) and
self-similar IFS (when the
fi
's are ane similarities). The notation dened at
the beginning of Section 4.3 are used.
Let us recall the denition of a weakly conformal map.
Denition 4.3.8.
Let
m≥2
be an integer,
U⊂Rd
an open set,
S={fi}m
i=1
where each
fi
is a
C1
contractions from
U
to
U
and
K
the attractor of
S
.
One says that
S
is weakly conformal when, recalling
(4.21)
,
lim
k→+∞
sup(xi)i∈N∈{1,...,m}Nlog qf′
(x1,...,xk)(π(σk(x))) q−log ||f′
(x1,...,xk)(π(σk(x)))||
k= 0.
(4.32)
In this case, a measure dened by
(4.22)
is called a weakly conformal-measure.
Example 4.3.9.

If the maps
f1, ..., fm
are ane similarities or confor-
mal maps (i.e verify
||f′(x)(y)|| =||f′(x)||·||y||
for every
x∈U, y ∈Rd
),
the system
S={f1, ..., fm}
is weakly conformal. In this case the IFS
is called self-similar or self-conformal and the measures satisfying
(4.22)
are called respectively, self-similar and self-conformal measures. Note
that this class of IFS contains for instance every system of holomorphic
contracting mappings.

Assume that for any
1≤i≤m, fi:Rd→Rd
is dened by
fi(x) =
Aix+bi,
where for any
1≤i≤m
,
bi∈Rd
and
Ai∈GLd(R)
has its
eigenvalues equal in modulus to
0< ri<1
and for any
1≤i, j ≤m
,
AiAj=AjAi
. Then
S={f1, ..., fm}
is weakly-conformal.
Let us recall important results about weakly-conformal measures.
4.3.2.1 Geometric properties of
C1
weakly-conformal IFS
Let
m≥2
be an integer. One collects some useful geometric results when
dealing with
C1
weakly conformal IFS.

4.3 ITERATED FUNCTION SYSTEMS 61
Consider
S={f1, ..., fm}
a
C1
weakly conformal IFS with attractor
K
and for every
x∈K
,
k∈N
and
i= (i1, .., ik)∈Λk
, write
ci(x) = ||f′
i(x)||.
Let us recall the following result established as [37, Lemma 5.4].
Lemma 4.3.10
([37])
.
For any
c > 1
, there exists a constant
D(c)>0
such
that, for every
k∈N
, for every
i∈Λk
and every
x, y ∈K
,
D(c)−1c−k||f′
i(x)|| · ||x−y|| ≤ ||fi(x)−fi(y)|| ≤ D(c)ck||f′
i(x)|| · ||x−y||
(4.33)
D(c)−1c−k||f′
i(x)|| ≤ |fi(K)| ≤ D(c)ck||f′
i(x)||.
(4.34)
Remark 4.3.11.
Let
X⊂U
be a compact set. It is proven in [37] that equation
(4.33)
actually holds for any
(x, y)∈X2.
Note that, for every
k∈N
and every
x∈K,
one has
c±k||f′
i(x)|| =||f′
i(x)||1+ ±klog c
log ||f′
i(x)|| .
Moreover, since there exists two constants
C1, C2>0
such that for every
1≤i≤m
and every
x∈K
,
C1≤qf′
i(x)q≤ ||f′
i(x)|| ≤ C2,
there also exists two constants
0< t1≤t2
such that
t1≤k
log ||f′
i(x)|| ≤t2.
Combining this fact with Lemma 4.3.10, for any
θ > 0
, there exists
e
Cθ>0
such that for every
k∈N
, every
i∈Λk
and every
x, y ∈K
,
e
C−1
θci(x)1+θ||x−y|| ≤ ||fi(x)−fi(y)|| ≤ e
Cθci(x)1−θ||x−y||.
(4.35)
In particular, there also exists
b
Cθ
such that for every
i∈Λ∗
and every
x∈K,
one has
b
C−1
θc1+θ
i(x)|K| ≤ |fi(K)| ≤ b
Cθc1−θ
i(x)|K|.
(4.36)

62 CHAPTER 4: COVERING LEMMAS, HAUSDORFF MEASURES
AND IFS
Let us remark that (4.36) also implies that there exist
0< α ≤β < 1
as well
as
Cα, Cβ>0
such that, for any
k∈N,
Cααk≤ |fi(K)| ≤ Cββk.
(4.37)
4.3.2.2 Lyapunov exponent of
C1
weakly conformal IFS's
Let
m≥2
and let us x a
C1
weakly conformal IFS,
S={f1, ..., fm}
with
attractor
K
.
Given
x= (xn)n∈N∈ΛN,
the following quantity, called Lyapunov ex-
ponent of
S
at
x
, denes a logarithmic shrink rate associated with IFS
S
at
x
.
Proposition 4.3.12
([37])
.
For any
x= (xn)n∈N,
the Lyapunov exponent of
S
at
x
, is (well) dened as
λ(x) = −lim
n→+∞
log |fx1◦... ◦fxn(K)|
n.
(4.38)
Moreover, for any probability vector
(p1, ..., pm)∈[0,1]m
, denoting
ν∈ M(ΛN)
the measure dened by
ν([i]) = pi,
then there exists
λν≥0
such that for
ν
-
almost any
x= (xn)n∈N
,
λ(x) = Zλ(y)dν(y) := λν.
(4.39)
Remark 4.3.13.

By
(4.37)
, the Lyapunov exponent are uniformly (with
respect to weakly conformal measures) bounded by above and below by
some positive constant.

When
S
is a self-similar and
0< c1, ..., cm<1
are the contracting ratio
associated with the similarities
f1, ..., fm,
the lyapunov exponent of
ν
as
in Proposition 4.3.12 is simply
λν=−X
1≤i≤m
pilog ci.
The following consequence of Proposition 4.3.12 will be useful later on
(see Proposition 10.3.2).
Corollary 4.3.14.
Let
((p(k)
1, ..., p(k)
m))k∈N∈([0,1]m)N
be a sequence of proba-
bility vectors such that
(p(k)
1, ..., p(k)
m)→(p1, ..., pm).
Denote for
k∈Nν, νk∈
M(ΛN)
the measures dened by, for any cylinder
[(i1, ...in)],
νk([(i1, ..., ik)]) = p(k)
i1·... ·p(k)
in
and
ν([(i1, ..., in)]) = pi1·... ·pin.

4.3 ITERATED FUNCTION SYSTEMS 63
Then
νk→
k→+∞ν
weakly, so that
lim
k→+∞λνk=λν.
4.3.2.3 Dimension of weakly-conformal IFS's
Let us recall the following fundamental result.
Theorem 4.3.15
(Feng-Hu [37])
.
Let
(p1, ..., pm)∈[0,1]m
be a probability
vector,
ν∈ M(ΛN)
dened by, for any
i∈Λ∗, ν([i]) = pi
and
µ=ν◦π−1.
There exists
h≥0
such that for
µ
-almost every
x∈K
, there exists
µπ−1({x})∈ M(ΛN)
such that:
(1)
µπ−1({x})(π−1({x})) = 1,
(2) for
µπ−1({x})
-almost
y= (y1, ..., yn, ..)
,
−log µπ−1({x})([y1, ..., yn])
n→h.
(4.40)
(3) for every Borel set
A⊂ΛN,
ν(A) = ZK
µπ−1({x})(A)dµ(x).
(4.41)

denoting
λ
the Lyapunov exponent associated with
ν
(
(4.39)
),
µ
is exact-
dimensional (Denition 4.14) and
dim(µ) = −h−P1≤i≤mpilog pi
λ.
In the rest of this sub-section, we dene the pressure function associated
with a weakly conformal IFS, which is naturally related with the dimension
of the attractor. The proof of the good denition of this quantity is very
standard and does not diverge much from the proof in the conformal case.
Those computations in the weakly conformal case does not seem to be made
explicitly in the literature and for the seek of completeness, the proof are made
in this manuscript.
Proposition 4.3.16.
Let
m≥2
be an integer,
S={f1, ..., fm}
be
C1
weakly
conformal IFS and
K
its attractor.
Let us x
s≥0
and
z∈K.
The following quantity is well dened and

64 CHAPTER 4: COVERING LEMMAS, HAUSDORFF MEASURES
AND IFS
independent of the choice of
z:
Pz(s) = lim
k→+∞
1
klog X
i∈Λk||f′
i(z)||s.
(4.42)
Proof.
Assume rst that the limit exists in
R∪ {−∞}
and let us show that it
is independent of the choice of
z
and that the limit is
>−∞
. Let
c > 1
be a a
real number. By (4.3.10), following the notation involved, for any
k∈N,
one
has
log ÑX
i∈Λk
D(c)−sc−sk|fi(K)|sé≤log ÑX
i∈Λk||f′
i(z)||sé≤log ÑX
i∈Λk
D(c)scsk|fi(K)|sé.
(4.43)
Since (4.43) holds for any
c > 1
, one gets that
lim
k→+∞
1
kÑlog ÑX
i∈Λk||f′
i(z)||sé−log ÑX
i∈Λk|fi(K)|séé= 0,
(4.44)
which proves that this quantity does not depend on
z
. Moreover, there exists
b > 0
so that for any
k∈N
, any
i∈Λk
, any
x∈K
,
||f′
i(x)|| ≥ bk.
This implies that if
Pz(s)
is well dened, then
Pz(s)>−∞.
Let us now prove that the limit exists. For
k∈N,
write
gk= log ÑX
i∈Λk|fi(K)|sé.
(4.45)
As in the conformal case, the existence of the pressure relies on a sub-additivity
argument.
Lemma 4.3.17.
For any
ε > 0,
there exists a constant
Mε>0
such that for
any
n, m ∈N,
one has
gn+m≤Mε+mε +gn+gm.
(4.46)
Furthermore, any sequence
(gn)n∈N
verifying
(4.46)
is such that
(gn
n)n∈N
con-
verges in
R∪ {−∞}
.
Proof.
Let us start by proving the second statement. Let
(gn)n∈N
be a sequence
satisfying (4.46). Fix
ε > 0
and
Mε
satisfying (4.46). For any
q∈N, b ∈N,

4.3 ITERATED FUNCTION SYSTEMS 65
0≤r < q
, one has
gbq+r≤bgq+gr+ (bq +r)ε+ (b+ 1)Mε,
⇒gbq+r
bq +r≤bq
bq +r·gq
q+(b+ 1)Mε+gr
bq +r+ε.
Fixing
q
large enough independently of
b
so that
(b+1)Mε
bq ≤ε
, for any large
b∈N
, one has
gbq+r
bq +r≤(1 + ε)gq
q+ 2ε.
This implies that
lim sup
n→+∞
gn
n≤(1 + ε) lim inf
n→+∞
gn
n+ 2ε.
Letting
ε→0
proves the statement.
One now shows that
gn
satises (4.46).
Let
k∈N
and
i∈Λk.
Let us begin by the following lemma.
Lemma 4.3.18.
Following the notation of Lemma 4.3.10, one has, for any
j∈Λ∗,
1
2D(c)−2c−2k|fi(K)|·|fj(K)| ≤ |fij (K))| ≤ 2D(c)2c2k|fi(K)|·|fj(K)|.
(4.47)
Proof.
Let us start by establishing the lower-bound.
Let
x, y ∈K
such that
||fj(x)−fj(y)|| ≤ |fj(K)| ≤ 2||fj(x)−fj(y)||.
(4.48)
By Lemma 4.3.10, one has
D(c)−1c−k||f′
i(fj(x))|| · ||fj(x)−fj(y)|| ≤ ||fij(x)−fij(y)|| ≤ |fij(K)|,
(4.49)
and
||f′
i(fj(x))|| ≥ D(c)−1c−k|fi(K)|.
(4.50)
Combining (4.48), (4.49) and (4.50), one obtains
1
2D(c)−2c−2k|fi(K)|·|fj(K)|≤|fij (K)|.
Let us focus now on the upper-bound. Let
x, y ∈K
such that
||fij(x)−fij(y)|| ≥ 1
2|fij(K)|.
(4.51)

66 CHAPTER 4: COVERING LEMMAS, HAUSDORFF MEASURES
AND IFS
Using again Lemma 4.3.10, one has
||fij(x)−fij(y)|| ≤ D(c)ck||f′
i(fj(x))|| · ||fj(x)−fj(y)|| ≤ D(c)2c2k|fi(K)|·|fj(K)|.
(4.52)
The upper-bound is obtained by combining (4.51) and (4.52).
By Lemma 4.3.18, for any
c > 1
and any
n, n′∈N,
one has
gn+n′= log ÑX
i∈Λn+n′|fi(K)|sé= log ÑX
i∈Λn,j∈Λn′|fij (K)|sé
≤log ÑX
i∈Λn,j∈Λn′
2sD(c)2sc2sn|fi(K)|s|fj(K)|sé
=n·2slog(c) + log(2sD(c)2s) + log Ñ(X
i∈Λn|fi(K)|s)×(X
j∈Λn′|fj(K)|s)é
≤2sn log(c) + log(2sD(c)2s) + gn+gn′.
Fixing
c=eε
2s
, one has
2slog(c) = ε
and setting
Mε= log(2sD(c)2s)
shows
that
(gn)n∈N
satises the condition of Lemma 4.3.17.
Lemma 4.3.17 together with (4.44) concludes the proof of Proposition
4.3.16.
Since
Pz(s)
does not depend on
z
, one writes
Pz(s) = P(s) = lim
k→+∞
1
klog ÑX
i∈Λk|fi(K)|sé.
As said above, the pressure function is naturally connected to the dimension
of the attractor
K
associated with the underlying IFS. More precisely, the
following quantity is a natural candidate for the dimension
K
.
Denition 4.3.19.
Let
m≥2
be an integer. Let
S={f1, ..., fm}
be
C1
weakly
conformal IFS and
K
its attractor.
Let us denote by
dim(S)
the unique solution to
P(s) = 0.
One says that
dim(S)
is the conformality dimension of
S
.

4.3 ITERATED FUNCTION SYSTEMS 67
The following notion was introduced by Barral and Feng in [4] and will
be usefull in this manuscript.
Denition 4.3.20
( [4])
.
One says that
S
is dimension regular if for any
weakly conformal measure
µ∈ M(Rd)
associated with the probability vector
(p1, ..., pm)∈[0,1]m
and
S
, recalling
(4.3.12)
and denoting
ν∈ M(ΛN)
verify-
ing
µ=ν◦π−1
, one has
dim(µ) = min ®−P1≤i≤mpilog(pi)
λν
, d´,
(4.53)
where
λν
is dened by
(4.39)
.
Remark 4.3.21.

When
S
is self-similar, calling
0< c1, ..., cm<1
the
contraction ration of the similarities
f1, ..., fm
, for any probability vector
(p1, ..., pm), µ
and
ν
as in Denition 4.3.20, one has
dim(µ) = min ®−P1≤i≤mpilog(pi)
λν
, d´= min ®P1≤i≤mpilog(pi)
Pm
i=1 pilog(ci), d´.
(4.54)

As proved in [43], any self-similar IFS on
R
satisfying ESC is dimension
regular.
In the next subsection, we collect some results about weakly-conformal
IFS satisfying AWSC. In particuler, we prove that if
S
is such an IFS and has
no exact overlaps, then
S
is dimension regular.
4.3.3 Weakly conformal systems satisfying AWSC
4.3.3.1 Dimension regularity of
C1
weakly conformal IFS without
exact overlaps
We will prove the following result.
Proposition 4.3.22
(D.)
.
Assume that
S={f1, ..., fm}
satises the AWSC
without exact overlaps. Then
S
is dimension regular and
dim(S) = dimH(K).
Had the IFS been conformal and satisfying some bounded distortion prop-
erties, the proof of Proposition 4.3.22 would follow directly from the existence
of appropriated Gibbs measures. Unfortunately, such measures does not always
exists in the weakly-conformal case but some measures that are close enough
from satisfying the desired properties still exist as established by the following
lemma.

68 CHAPTER 4: COVERING LEMMAS, HAUSDORFF MEASURES
AND IFS
Lemma 4.3.23.
Let
ε > 0
and
s≥0
be a real numbers. There exists
k∈N
and
a probability vector
(pi)i∈Λk
such that the weakly conformal measure
ν
associated
with
S′={fi}i∈Λk
and
(pi)i∈Λk
veries, for any
p∈N
and
i1, ..., ip∈Λk,
e−kpε |fi1...ip(K)|s
epkP (s)≤ν([i1...ip]) ≤ekpε |fi1...ip(K)|s
epkP (s)
(4.55)
Proof.
Fix
ε > 0
and
c > 1
small enough so that
8slog c≤ε.
By Lemma 4.3.17 there exists
k∈N
so large that the constant named
D(c)
in Lemma 4.3.10 veries
log D(c)
k≤log c
and
|1
klog X
i∈Λk|fi(K)|s−P(s)| ≤ ε
2.
(4.56)
Writing again
gk= log Pi∈Λk|fi(K)|s,
let us dene the probability vector
(pi)i∈Λk
by setting
pi=|fi(K)|s
egk.
Let
ν
be the weakly conformal measure associated with
S′={fi}i∈Λk
and
(pi)i∈Λk.
Applying Lemma 4.47, for any
p∈N, i1, ..., ip∈Λk,
,
D(c)−2pc−2kp ≤|fi1◦... ◦fip(K)|
Qp
j=1 |fij(K)|≤D(c)2pc2kp.
(4.57)
Also
D(c)2spc2skp =epk2s·(log D(c)
k+log c)≤eε
2pk.
(4.58)
As a consequence, for any
p∈N
and any
i1, ..., ip∈Λk,
one has
ν([i1...ip]) = pi1·... ·pip=Qp
j=1 |fij(K)|s
epgk=Qp
j=1 |fij(K)|s
ekp(gk
k−P(s))epkP (s).
Using (4.56), (4.57) and (4.58) concludes the proof.
Remark 4.3.24.
The measure
ν
can be extended over
ΛN
by the usual argu-
ments. Moreover, for any
i= (i1, ..., in)∈Λ∗,
write
n1=k⌊n
k⌋
and
n2=
k(⌊n
k⌋+ 1).
Consider
j∈Λn1
such that
[i]⊂[j]
and
ℓ= (ℓ1, ..., ℓn2−n)∈Λn2−n,
one has
e−n2ε|f(i1,...,in,ℓ1,...,ℓn2−n)(K)|s
en2P(s)≤ν([iℓ]) ≤ν([i]) ≤ν([j]) ≤en1ε|f(i1,...,in1)(K)|s
en1P(s).
(4.59)
By
(4.47)
, there exists a constant
C > 0
such that, uniformly on
i, j, iℓ,
one

4.3 ITERATED FUNCTION SYSTEMS 69
has
C−1≤min ®|fj(K)|
|fi(K)|,|fi(K)|
|fiℓ(K)|´≤max ®|fj(K)|
|fi(K)|,|fi(K)|
|fiℓ(K)|´≤C.
Hence there exists a constant
γs,ε
such that for any
i= (i1, ..., in)∈Λ∗
, one
has
γ−1
s,ε e−nε |fi(K)|s
enP (s)≤ν([i]) ≤γs,εenε |fi(K)|s
enP (s).
(4.60)
Let us now prove Proposition 4.3.22.
Proof.
Call
K
the attractor of
S
. Let us show rst that if any system
S
satisfying the AWSC also veries that, for any weakly-conformal measure
µ∈
M(Rd)
associated with a probability vector
(p1, ..., pm)
and
S
,
dim(µ) = −P1≤i≤mpilog pi
λν
(4.61)
where
ν
is the measure associated on
ΛN,
then
dim(S) = dimH(K).
Fix
ε > 0
consider
k∈N
,
S′={fi}i∈Λk
and
ν
as in Lemma 4.3.23 applied
with
s= dim(S).
Note that, since
S
satises the AWSC, so does
S′
. Then,
considering the measure
µ=ν◦π−1
, where
π
is the canonical projection, one
has
dim(S)−ε≤dim(µ) = −Pi∈Λkpilog pi
λν≤dim(S) + ε.
This proves that
dimH(K)≥dim(S)−ε.
Since it always holds that
dimH(K)≤
dim(S)
(see [30]) and
ε
is arbitrary,
dimH(K) = dim(S).
Let us show that, for any system satisfying the AWSC, (4.61) holds for
every weakly conformal measure
µ
.
Let
µ∈ M(Rd)
be a weakly conformal measure associated with
S
and a
probability vector
(p1, ..., pm)
and
ν∈ M(ΛN)
such that
µ=ν◦π−1.
It comes from from the proof of Theorem 4.3.15 [37] (applied to
µ
), that
for any
ε > 0,
for
µ
-almost any
x∈K
such that
µπ−1({x})
exists and satises
the two rst items of Theorem 4.3.15, there exists
n0
large enough so that, for
any
n≥n0
, there exists
i1, ..., iNn
such that:

for any
1≤j≤Nn,
e−n(λ+ε)≤ |fij(K)| ≤ e−n(λ−ε),
(4.62)

70 CHAPTER 4: COVERING LEMMAS, HAUSDORFF MEASURES
AND IFS

one has
µπ−1({x}) [
1≤j≤Nn
[ij]!≥1
2,
(4.63)

for any
1≤j≤Nn,
e−n(h+ε)≤µπ−1({x})([ij]) ≤e−n(h−ε)
(4.64)
Assume that
h > 0
and take
0<ε<min h
2,λ
2.
Combining (4.63) and (4.64), one gets
Nn≥1
2en(h−ε).
(4.65)
Note that
#k:e−n(λ+ε)≤2−k≤e−n(λ−ε)≤2nε
log 2 .
As a consequence, there
exists
k∈[n(λ−ε)
log 2 ,n(λ+ε)
log 2 ]
such that
#Λ(k)∩[ij]1≤j≤Nn≥Nn
2nε
log 2 ≥
1
2enh
2
2nε
log 2
.
(4.66)
Since for any
≤j≤Nn,[ij]∩π−1({x})=∅,
one also has
fij(K)⊂B(x, e−n(λ−ε)),
so that, writing
n′=⌊n(λ−ε)
log 2 ⌋
, one has
#¶i∈Λ(n′):fi(K)∩B(x, 2−n′)©≥
1
2enh
2
2nε
log 2
.
(4.67)
In particular, recalling (4.28) and Propostion 4.3.28,
log tk
k↛0
and
S
does not satisfy the AWSC. As a consequence,
S
satises the AWSC
implies
h= 0
, which, recalling the last item of Theorem 4.3.15, concludes the
proof.
Remark 4.3.25.
If
S
is a self-similar system and satises the OSC, then it
satises the AWSC and has no exact overlaps, so that Proposition 4.3.22 holds
for
S
.
We concludes this sub-section by giving some examples of self-similar
measures satisfying AWSC with no exact overlaps. These examples are based
on the following result proved by Barral and Feng in [4].
Theorem 4.3.26
([4])
.
Let
S
be a self-similar IFS on
R
satisfying the expo-
nential separation condition (see Denition 4.3.3). Then
E
satises the AWSC
if and only if
dim(S)≤1.

4.3 ITERATED FUNCTION SYSTEMS 71
Corollary 4.3.27.

Let
m∈N
and
c1, ..., cm∈(0,1)m
satisfying
c1+... +cm≤1.
Then by the proof of [43, Theorem 1.8] and Theorem 4.3.26 , for
Lm
almost all
(a1, ..., am)∈Rm
, setting
f1(x) = c1x+a1, ..., fm(x) = cmx+am,
the system
S={f1, ..., fm}
satises the AWSC, hence Theorem 11.1.4
applies to any self-similar measures associated with
S
.

One can also provide explicit examples: set
f1(x) = 1
4x, f2(x) = 1
4(x+ 1)
and
f3(x) = 1
4(x+t)
where
t
is irrational. Then, the same proof as in [43,
Theorem 1.6] and Theorem 4.3.26 yields that
S={f1, f2, f3}
satises the
AWSC and has no exact overlaps.

It is worth mentioning that the proof of Theorem 4.3.26 relies on [68, The-
orem 6.6] which is currently established only on
R.
An higher dimension
version of this result would most likely lead to more examples.
In the following section, we prove that the AWSC can be equivalently
dened using the sets
Λ(k)
and
e
Λ(k)
(see (4.25)).
4.3.3.2 Equivalent denitions of AWSC
For
k∈N
and
x∈Rd,
recalling (4.26) and (4.25), set
Tk(x) = ¶fi:fi(K)∩B(x, 2−k)=∅, i ∈e
Λ(k)©
T′
k(x) = ¶fi:fi(K)∩B(x, 2−k)=∅, i ∈Λ(k)©.
Note that
S
satises AWSC
⇔limk→+∞maxx∈Rdlog Tk(x)
k= 0.
Proposition 4.3.28.
One has
lim
k→+∞max
x∈Rd
log Tk(x)
k= 0 ⇔lim
k→+∞max
x∈Rd
log T′
k(x)
k= 0.
Proof.
By (4.37), there exists
0< α < 1
2< β ≤1
such that for every
k∈N
,
αk≤ |fi(K)| ≤ βk.
Remark 4.3.29.
(1): For every
k∈N
and every
i= (i1, ..., in)∈Λ(k),
one
has
C(α, β)−1k≤k−log 2
log β+ 1 ≤n≤2k−log 2
log α≤C(α, β)k.

72 CHAPTER 4: COVERING LEMMAS, HAUSDORFF MEASURES
AND IFS
(2): For every
c > 1
, by Lemma 4.47, for every
i= (i1, ..., in)∈e
Λ(k)
,
D(c)−2min
1≤j≤m|fj(K)|c−2C(α,β)k2−k≤ |fi(K)| ≤ 2−k.
In particular, for any
k∈N
large enough, one has
c−1
2C(α,β)k2−k≤ |fi(K)| ≤ 2−k.
(4.68)
Lemma 4.3.30.
For every
ε0>0
, there exists
kε∈N
such that for every
k≥kε0
, for every
i= (i1, ..., in)∈Λ(k),
there exists
0≤p≤ε0k
such that
(i1, ..., in−p)∈e
Λ(k).
Proof.
Fix
ε=ε0
2C(α,β)
and
c > 1
such that
c1−εβε<1
. By Lemma 4.47, for
any
(i1, ..., in)∈Λ∗
and
0≤p≤n,
|f(i1,...,in)(K)| ≤ D(c)2cn−p|f(i1,...,in−p)(K)|×|f(in−p+1 ,...,in)(K)|.
In particular, for
p≥nε,
|f(i1,...,in−p)(K)| ≥ D(c)−2c−(n−p)
|f(in−p+1,...,in)(K)|2−k≥2−kD(c)−2
(c1−εβε)n.
This yields, for
k
large enough and
p=⌊εC(α, β)k⌋+ 1 ≤2C(α, β)εk
that
|f(i1,...,in−p)(K)|>2−k.
As a consequence, there must exists
p≤2C(α, β)εk
such that
(i1, ..., in−p)∈
e
Λ(k).
Lemma 4.3.31.
For every
c > 1
, for every
ε > 0,
for every
k
large enough
(depending on
c
) and every
x∈Rd,
one has
#Tk(x)≤k⌊
C(α,β)
2log c
log 2 ⌋Cdckd C(α,β )
2max
k≤k”≤k(1+⌊
C(α,β)
2log c
log 2 ⌋)
max
y∈Rd#T′
k′(y)
(4.69)
and
#T′
k(x)≤mkε#Tk(x).
Proof.
Remark that, for each
k′
such that
c−1
2C(α,β)k2−k≤2−k′≤2−k,
there exists a dimensional constant
Cd
so that each ball
B(x, 2−k)
can be cov-
ered by less than
Cdckd C(α,β)
2

4.3 ITERATED FUNCTION SYSTEMS 73
balls of radius
2−k′.
This implies that
#¶fi, i ∈Tk(x)∩Λ(k′)©≤Cdckd C(α,β )
2max
y∈Rd#T′
k′(y).
Since one has
e
Λ(k)⊂
k(1+⌊
C(α,β)
2log c
log Z⌋)
[
k′=k
Λ(k′),
it holds that
#Tk(x)≤k⌊
C(α,β)
2log c
log 2 ⌋Cdckd C(α,β )
2max
k≤k”≤k(1+⌊
C(α,β)
2log c
log 2 ⌋)
max
y∈Rd#T′
k′(y).
(4.70)
Moreover, by Lemma 4.3.30, there exists
ϕk:T′
k(x)→Tk(x)
dened by
ϕk((i1, ..., in)) = (i1, ...in−p)
with
0≤p≤kε
. The mapping
ϕk
veries that each ber has cardinality
smaller than
mkε
. This implies that
#T′
k(x)≤mkε#Tk(x).
Taking the log of the estimates of Lemma 4.3.31 and letting
k
tends to
innity concludes the proof.
4.3.4 Dimension properties of self-similar measures
In this section, one recalls what is known about the dimension theory of self-
similar measures and sets and the multifractal analysis of these measures. First
we start with some preliminaries.
4.3.4.1 Some preliminaries
Let
m≥2
and
S={f1, ..., fm}
be a self-similar system. It will be particularly
convenient here to adopt a slightly dierent denition of the sets
e
Λ(k)
and
Λ(k).
Start by setting
β= max
1≤i≤mci
and
θ= min
i=1≤mci.
(4.71)

74 CHAPTER 4: COVERING LEMMAS, HAUSDORFF MEASURES
AND IFS
In this section we set for every
k∈N
Λ(k)=i= (i1, ..., in)∈Λ∗:θk+1 < ci≤θk
(4.72)
e
Λ(k)=i= (i1, ..., in)∈Λ∗:ci≤θk< c(i1,...,in−1).
Let us collect few observations.
Remark 4.3.32.
(1): Its is easily veried that one can equivalently use the
new sets
Λ(k)
or
e
Λ(k)
in the denition of AWSC (see Denition 4.3.3 and
Proposition 4.3.28).
(2): For every
i= (i1, ..., in)∈e
Λ(k),
one has
θk≥ci=c(i1,...,in−1)·cn> θkcn≥θk+1
In particular, one has
e
Λ(k)⊂Λ(k).
(3):
Λ(k)k∈N
is a partition of
Λ∗.
(4): By iterating the self-similarity equation
(4.22)
, for every
k∈N,
one has
µ(·) = X
i∈e
Λ(k)
piµ(f−1
i(·)).
(5): For every
i∈Λ(k)
a similar argument as in Proposition 4.3.28 yields the
existence of an integer
0≤p≤2⌊log θ
log β⌋
such that
(i1, ..., in−p)∈e
Λ(k).
Lemma 4.3.33.
Let
r > 0
be a real number and
x∈Rd.
For every
k≥
⌊log r
log θ⌋+ 2
, one has
µ(B(x, r)) ≤X
i∈e
Λ(k):fi(K)∩B(x,r)=∅
pi≤µ(B(x, 2r)).
(4.73)
Moreover,
1≤Pi∈Λ(k):fi(K)∩B(x,r)=∅pi
Pi∈e
Λ(k):fi(K)∩B(x,r)=∅pi≤m2⌊log θ
log β⌋.
(4.74)
Proof.
Fix
k≥ ⌊log r
log θ⌋+ 2.
Note that for any
i∈e
Λ(k)
, if
fi(K)∩B(x, r) = ∅,
then
µ(f−1
i(B(x, r))) = µ(K∩f−1
i(B(x, r))) = µ(f−1
i(fi(K)∩B(x, r)))
=µ(∅) = 0.

4.3 ITERATED FUNCTION SYSTEMS 75
Also if
fi(K)⊂B(x, r),
1≥µ(f−1
i(B(x, r))) ≥µ(f−1
i(fi(K))) = µ(K) = 1.
Using item
(4)
of Remark 4.3.32 and noticing that for every
i∈Λ(k)
,
fi(K)⊂
B(x, 2r)
yields equation (4.73).
Let us now prove that (4.74) holds. The left-hand side is a straightforward
consequence of item
(2)
of Remark 4.3.32.
By item
(5)
of Remark 4.3.32, at each
i= (i1, ..., in)∈Λ(k)
corresponds
(i1, ..., in−p)∈e
Λ(k)
with
p≤2⌊log θ
log β⌋
. Also, one has
pi=p(i1,...,in−p)×p(in−p+1,...,in)≤p(i1,...,in−p).
In addition, at each such word
(i1, ..., in−p)∈e
Λ(k)
can correspond at most
m2⌊log θ
log β⌋
words
i= (i1, ..., in)∈Λ(k).
This yields the right-hand side.
In particular, for very ball
B,
and
k≥ ⌊log( 1
2|B|)
log(β)⌋+ 1
one has
µ(B)≤X
i∈Λ(k)
β,fi(K)∩B=∅
pi≤µ(2B).
(4.75)
4.3.4.2 Dimension of self-similar measures
Let us start with the following remark.
Remark 4.3.34.
If the mappings
f1, ..., fm
are ane similarities, then the
conformality dimension is called the similarity dimension. It is the unique real
number
s
solution to
m
X
i=1
cs
i= 1.
(4.76)
The following Theorem summarized some of the known results about the
dimension of self-similar measures and the dimension of the attractor associated
with those self-similar systems. In the case where the IFS satises the OSC,
the reader may refer to [32], the case where the IFS satises the AWSC is
established in this manuscript in Proposition 4.3.22 and in the case where the
IFS satises the ESC, one refers to [43] for more details.
Theorem 4.3.35.
Let
S={f1, ..., fm}
be a self-similar IFS. Denote by
K
its
attractor and
0< c1, ..., cm<1
the contracting ratios of
f1, ..., fm.
1. Assume that
S
satises the AWSC and has no exact overlaps (or the
OSC), i.e, for any
i=j∈Λ∗, fi=fj
. Then
dimH(K) = dim(S)

76 CHAPTER 4: COVERING LEMMAS, HAUSDORFF MEASURES
AND IFS
and, for any self-similar measure associated with
S
and any probability
vector
(p1, ..., pm),
one has
dim(µ) = P1≤i≤mpilog(pi)
P1≤i≤mpilog(ci).
2. Assume that
S
satises the ESC and that no non trivial linear subspace
V⊂Rd
is preserved by the linear part of each
fi.
Then
dimH(K) = min {dim(S), d}
and for any self-similar measure
µ
associated with
S
and any probability
vector
(p1, ..., pm),
one has
dim(µ) = min ®P1≤i≤mpilog(pi)
P1≤i≤mpilog(ci), d´.
Item
(2)
shows that IFS's on
R
verifying the ESC actually fall in the more
general category of dimension regular self-similar IFS's (Denition 4.3.20).
4.3.4.3 Multifractal analysis of self-similar measures
Let us recall what is known in general about the multifractal formalism of
self-similar measures.
Proposition 4.3.36.
Let
µ
be a self-similar measure. Then:
1.
τµ
is dierentiable at
q= 1+
and
τ′
µ(1+) = dimH(µ),
[69, Theorem 5.1],
2.
τµ(0) = −dimH(supp(µ)),
3. [35] For any
q > 0,
τµ(q) = lim inf
n→+∞−log Θµ(q, n)
n= lim
n→+∞−log Θµ(q, n)
n.
4. [35] At any
q≥1
for which
τ′
µ(q)
exists (which is the case outside an
at most countable set of
q
's),
µ
satises the multifractal formalism at
h=τ′
µ(q),
so that
τ∗
µ(h) = dimH(Eh
µ) = qh −τµ(q).
(4.77)
Let us also mention that it is known that some self-similar measures do
not satisfy the multifractal formalism for
q < 0
 [72] (i.e at
h > τ′
µ(0+)
).

4.3 ITERATED FUNCTION SYSTEMS 77
The following quantity plays a particular role in the multifractal analysis
of self-similar measures.
Denition 4.3.37.
Let
(p1, ..., pm)∈(0,1)m
be a probability vector and
µ∈
M(Rd)
the self-similar associated with
S
(i.e, satisfying
(4.22)
). For
q∈R
,
let us dene
T(q)
as the unique solution to
m
X
i=1
pq
ir−T(q)
i= 1.
(4.78)
Proposition 4.3.38
([32])
.

q7→ T(q)
is a concave real analytic mapping
and in particular, dierentiable over
R
. Moreover, the equation
T′′(q) = 0
admits only a nite number of solutions so that
T
is strictly concave.

T(0) = −dim(S).

T′(1) = P1≤i≤mpilog(pi)
P1≤i≤mpilog(ri).
One summarizes here what is known about the multifractal analysis of
self-similar measures depending on which separation condition they verify. One
also compute the
Lq
scaling function of self-similar measures satisfying AWSC
with no exact overlaps. Note that, on
R,
this class contains the class of self-
similar measures associated with systems satisfying the ESC and with similarity
dimension smaller than one.
Theorem 4.3.39.
Let
µ∈ M(Rd)
be a self-similar measure associated with
S
.
1. [32] Assume that
S
satises the OSC, then one has
τµ(q) = T(q).
(4.79)
Moreover, for any
h∈[τ′(+∞), τ ′(−∞)]
, writing
h=τ′(q),
dimH(‹
Eh
µ) = dimH(Eh
µ) = τ∗
µ(h) = qT ′(q)−T(q).
In particular,
µ
satises the mulfractal formalism at any
h≥0
(see
Denition 4.2.14).
2. Assume that
S
satises the AWSC, then for any
q > 0
,
µ
satises the
multifractal formalism at any
h∈(τ′
µ(+∞), τ ′
µ(0+))
[36] . Moreover, if
µ
has no exact overlaps, then
τµ(q) = T(q)
for any
q > 0
.
3. [4] Assume that
S
is dimension regular, then
τµ
is dierentiable on
(0,1]
and
µ
satises the multifractal formalism at any
h∈[τ′
µ(1), τ ′
µ(0+)].
More-
over,
τµ
is given as follows:

78 CHAPTER 4: COVERING LEMMAS, HAUSDORFF MEASURES
AND IFS

If
T′(1) > d,
then, for all
q∈[0,1], τµ(q) = d(q−1).

If
T′(1) < d
and
T(0) ≥ −d
then
τµ(q) = T(q).

If
T′(d)> d
and
T(0) <−d,
then setting
eq= inf {q:qT ′(q)−T(q)≤d},
one has



If
q∈[0,eq), τµ(q) = q(d+T(eq))
eq−d
If
q∈[eq, 1], τµ(q) = T(q).
(4.80)
4. [68] If
d= 1
and
S
satises the ESC, then for any
q≥1,
τµ(q) = min {q−1, T (q)}.
In particular,
τµ
is dierentiable at any
q=q0
, where
q0
is dened (if
possible) by
q0−1 = T(q0).
As a consequence,
τµ
satises the multifractal
formalism at any
h∈[τ′(+∞), τ ′(0+)] \(τ′
µ(q0−), τ ′
µ(q0+)).
Proof.
Let us show the second part of Item
(2)
. Assume that
S
satises the
AWSC and has no exact overlaps.
Let
q > 0
,
k∈N
and
D∈ Dk
. On one hand, recalling (4.28) and item
(4)
of Remark 4.3.32,
µ(D)q≤ÑX
i∈e
Λ(k):fi(K)∩D=∅
piéq
≤tq
kmax
i∈e
Λ(k):fi(K)∩D=∅
pq
i
(4.81)
≤tq
kX
i∈e
Λ(k):fi(K)∩D=∅
pq
i.
(4.82)
On the other hand,
ÑX
D′∈Dk:D′∩D=∅
µ(D′)éq
≥ÑX
i∈e
Λ(k):fi(K)∩D=∅
piéq
≥max
e
Λ(k):fi(K)∩D=∅
pq
i
(4.83)
≥t−1
kX
i∈e
Λ(k):fi(K)∩D=∅
pq
i.
(4.84)
Note also that
µ(D)q≤ÑX
D′∈Dk,D′∩D=∅
µ(D′)éq
≤2dq max
D′∈Dk,D′∩D=∅
µ(D′)q
(4.85)
≤2dq X
D′∈Dk,D′∩D=∅
µ(D′)q
(4.86)

4.3 ITERATED FUNCTION SYSTEMS 79
and
X
D∈Dk
µ(D)q≤X
D∈DkX
D′∈Dk:D′∩D=∅
µ(D′)q≤2dX
D∈Dk
µ(D)q.
(4.87)
Let us also remark, that, since, for any
D=‹
D∈ Dk
,
|D|=|‹
D|= 2−k
and
D∩‹
D=∅,
by Lemma 4.1.4, for any
i∈e
Λ(k)
,
fi(K)
intersects at most
γd
dyadic
cubes of generation
k
.
Combining this remark with (4.81),(4.83), (4.85) and (4.87), one gets
2−q(d+1)t−1
kX
i∈e
Λ(k)
pq
i≤X
D∈Dk
µ(D)q≤tq
kγdX
i∈e
Λ(k)
pq
i.
(4.88)
By denition of
T(q),
2T(q)(−k−1)
|K|≤X
i∈e
Λ(k)
pq
i=X
i∈e
Λ(k)
cT(q)
ic−T(q)
ipq
i≤2−T(q)k|K|.
(4.89)
Recalling that
limk→+∞log tk
k= 0
, (4.88) and (4.89), one obtains
lim
k→+∞
log(PD∈Dkµ(D)q)
−k=T(q).
(4.90)
Now that we recall the standard notion required to our study, we start
by giving an instructive construction.

80 CHAPTER 4: COVERING LEMMAS, HAUSDORFF MEASURES
AND IFS

Chapter 5
Anisotropic ubiquity for
quasi-Bernoulli measures
In this chapter, one presents a construction which mixes some of the approach
mentioned in introduction. Consider a sequence of balls
(Bn)n∈N.
The main
theorem of this chapter, Theorem 5.1.5, gives a lower-bound for the Hausdor
dimension of limsup sets generated by rectangles very much as in [74], in which
the measure they consider is the Lebesgue measure, but we assume here that
µ(lim sup]n→+∞Bn) = 1
where
µ
is a geometrical realization of a quasi-Bernoulli
measure.
Although this approach only deals with a particular case (in terms of
measure and shape of shrunk sets), we will see that some important geometric
quantities already appears naturally.
The rst Section, Section 5.1 states the main result, Theorem 5.1.5 and
Section 5.2 is dedicated to the proof of Theorem 5.1.5.
5.1 Anisotropique ubiquity and geometric quasi-
Bernoulli measures
As mentioned at the beginning of this chapter, a case one can treat to help us
understanding the geometric quantities involved when one deals with heteroge-
neous and anisoptropic mass transference principles, is the specic case where
the sequence
(Bn)n∈N
veries
µlim supn→+∞Bn= 1
and
µ
is a geometric
realization on the dyadic cubes of a quasi-Bernoulli measure fully supported
on
[0,1]d
. In this case, it is possible, given
1≤τ1≤... ≤τd
, to establish a
lower-bound for
lim supn→+∞Rn
, where
Rn⊂Bn
is a rectangle of length-sides
Qd
i=1 |Bn|τi.

82
CHAPTER 5: ANISOTROPIC UBIQUITY FOR QUASI-BERNOULLI
MEASURES
Let us start by recalling what a geometric realization of a quasi-Bernoulli
measure is.
Denition 5.1.1.
Let
µ∈ P([0,1]d)
. For
D∈ B([0,1]d)
such that
µ(D)>0
,
dene
µD=µ|D
µ(D).
When
D
is a closed dyadic subcube of
[0,1]d
,
TD:D→[0,1]d
stands
for the canonical ane mapping which sends
D
onto
[0,1]d
. In addition, when
µ(D)>0
one denes
µD=TDµD∈ P([0,1]d).
The measure
µ
is said to be quasi-Bernoulli when there exists a constant
Cµ≥1
such that for every
p∈N
and every
D∈ Dp
with
µ(D)>0
, one has
1
Cµ
µ≤µD≤Cµµ.
(5.1)
The measure
µD
is the renormalized restriction of
µ
to
D
and
µD
is the
rescaled version of
µD
on the unit cube.
Exemple 5.1.2.
Dene
Λ = {0,1}
,
Σ=ΛN
,
σ
be the shift operator on
Σ
, and
endow
Σ
with the standard ultra-metric distance. Let
π
the canonical projection
of
Σ
onto
[0,1]
. For any Hölder potential
ϕ
on
Σ
, denote by
νϕ
the unique
equilibrium state associated with
ϕ
on
Σ
(see [14]). Then the measure
µϕ=
νϕ◦π−1
is quasi-Bernoulli, and
νϕ
is also called a Gibbs measure associated with
φ
. This follows from the fact that there exists a number
P(φ)
, the topological
pressure of
φ
, and
C≥1
, such that for all
x∈Σ
, for all
n∈N
:
C−1≤νϕ({y= (yi)∞
i=1 ∈Σ : yi=xi
for all
1≤i≤n})
e−nP (φ)+Pn−1
k=0 φ(σkx)≤C.
Note that there exist quasi-Bernoulli measures obtained as projections of mea-
sures of Gibbs type associated to potentials
ϕ
with much weaker regularity prop-
erties (see [19,73]).
Remark 5.1.3.
It is easily seen that a quasi-Bernoulli measure
µ
, if not
supported on an ane hyperplane, is such that
µ(∂[0,1]d) = 0
. For oth-
erwise its orthogonal projection onto at least one of the sets
{0}i×[0,1] ×
{0}d−i−1
, which is quasi-Bernoulli as well, would have an atom at
(0,...,0)
or
(0,...,0
| {z }
i
,1,0,...,0
| {z }
d−i−1
)
. This should imply that it is a Dirac mass, hence
µ
is
supported on a hyperplane. This property will be used in the proof of our main
result.

5.1 ANISOTROPIQUE UBIQUITY AND GEOMETRIC
QUASI-BERNOULLI MEASURES 83
Let us recall the following result.
Proposition 5.1.4
([41])
.
A quasi-Bernoulli probability measure is exact di-
mensional.
One establishes the following ubiquity theorem:
Theorem 5.1.5
(D. [22])
.
Let
µ∈ P([0,1]d)
be a quasi-Bernoulli probability
measure fully supported on
[0,1]d
. Let
(Bn:= B(xn, rn))n∈N
be a sequence of
balls in
[0,1]d
such that
limn→+∞rn= 0
and
µ(lim supn→+∞Bn)=1.
Let
1≤τ1≤... ≤τd
be
d
real numbers,
τ= (τ1, . . . , τd)
and
(On)n∈N∈
Od(R)N
be a sequence of orthogonal matrices. For
n∈N
, set
Rn=xn+One
Rn,
where
e
Rn= diag(rτ1
n, ..., rτd
n)·[0,1]d
(5.2)
and
s(µ, τ) = min
1≤k≤dÇdim(µ) + P1≤j≤kτk−τj
τkå.
(5.3)
One has
dimH(lim sup
n→+∞
Rn)≥s(µ, τ).
(5.4)
Remark 5.1.6.
(1) For convenience, in particular to follow the point of view
adopted in [75], the results are stated with
Rd
endowed with
∥ ∥∞
and for
balls shrunk into rectangles with one vertex equal to the center of the shrunk
ball. However, we emphasize that, up to very slight modications of the proof
(essentially by adding constants at some places), they still hold for another
norm and if the balls are shrunk into rectangles containing the center of the
initial cube.
(2) Given
τ > 1
, by taking
τi=τ
for all
1≤i≤d
and
On=Id
for all
n∈N
, , Theorem 5.1.5 reduces to Barral-Seuret's theorem [6] in the special
case of quasi-Bernoulli measures.
(3) By taking
µ=Ld
and
On=Id
for all
n∈N
, we recover the result
established in [75], i.e., formula established in [74] in the case of the Lebesgue
measure.
Remark 5.1.7.

The proof does not entirely use the exact dimensionality of
µ
, the key property
is the quasi-Bernoulli property 5.1. However, the fact that
dimH(µ) = dimP(µ)
can be used to prove
dimH(lim supn→+∞Rn)≤s(µ, τ)
under additional as-
sumptions. The existence of upper bounds for the Hausdor dimension of lim-
sup sets (including of rectangles) included in balls
(Bn)n≥N
will be achieved in
chapter 8, under more general settings (see [6] for the case
τ1=··· =τd
).

84
CHAPTER 5: ANISOTROPIC UBIQUITY FOR QUASI-BERNOULLI
MEASURES

The expression of the lower-bound
(5.3)
is in fact naturally connected to the
Hausdor content
(4.7)
of rectangles (which is also, up to some multiplicative
constant, the so-called singular value function associated with
τ
, [51]). More
precisely,
s(µ, τ )
is the largest
s≥0
such that, for any
n∈N
,
Hs
Rn≥ |Bn|dim(µ).
This key remark will be exploited to deal with more general ubiquity theorems.
5.2 Proof of Theorem 5.1.5
Fix once and for all the quasi-Bernoulli measure
µ
,
1≤τ1≤... ≤τd
and
τ= (τ1, ..., τd)
. Recall that
α= dim(µ)
is the dimension of
µ
.
The lower bound of Theorem 5.1.5 will be obtained by constructing a
Cantor set included in
lim supn→+∞Rn
, and of dimension larger than or equal
to
s(µ, τ)
. Before starting the construction, the following helpful result is
recalled.
Lemma 5.2.1
([11])
.
Let
A=B(x, r)
and
B=B(x′, r′)
be two closed balls,
and
q≥3
be such that
A∩B=∅
and
A\(qB)=∅
. Then
r′≤r
and
qB ⊂5A.
Proof.
Consider
z∈A\qB
. One has
qr′≤ ∥z−x′∥∞≤ ∥z−x∥∞+∥x−x′∥∞≤r+r+r′.
Hence
q−1
2r′≤r
, and in particular, one necessarily has
r′≤r
and
qr′≤
2r+r′≤3r.
Furthermore, if
y∈qB
, then
∥y−x∥∞≤ ∥x′−y∥∞+∥x′−x∥∞≤qr′+r′+r≤5r.
This concludes the proof.
We construct thereafter a Cantor set
K
as well as a sequence of strictly
positive real numbers
(εp)p∈N
and a Borel probability measure
η
such that:

K⊂lim supn→+∞Rn
and
η(K) = 1
,

The sequence
(εp)p∈N
is decreasing with
limp→+∞εp= 0
and there exists
a constant
C
such that for any
p∈N
, there exists
rp>0
verifying, for
any ball
B⊂Rd
of radius
r
less than
rp
,
η(B)≤C.rs(µ,τ)−4εp.
(5.5)

5.2 PROOF OF THEOREM 5.1.5 85
Then, applying the mass distribution principle (Lemma 4.2.4), since
η(K) = 1
one deduces that, for any
p > 0
,
dimH(lim sup
n→+∞
Rn)≥dimH(K)≥s(µ, τ)−4εp,
hence, letting
p→+∞
will conclude.
The construction of
(K, η)
is decomposed into several steps. Without loss
of generality we assume that
s(µ, τ)>0
. Fix a decreasing sequence
(εp)p∈N
converging to 0 at
∞
, such that
ε0≤max(1, s(µ, τ)/4)
.
Step 1: Initialization
Let us start with a denition.
Denition 5.2.2.
For
ν∈ P([0,1]d)
,
β≥0
, and
ε, ρ > 0
, dene
Eβ,ε,ρ
ν=x∈[0,1]d:∀0< r ≤ρ, B(x, r)⊂[0,1]d
and
ν(B(x, r)) ≤rβ−ε.
Then set
Eβ,ε
ν=[
n≥1
Eβ,ε, 1
n
ν.
With
β=α= dimH(µ)
, since
µ(∂[0,1]d) = 0
(due to Remark 5.1.3 and
the assumption that
µ
is fully supported), for all
ε > 0
, one has
µ(Eα,ε
µ) = 1
.
For all
p∈N
, consider
ρp∈(0,1)
small enough so that
µ(Eα,εp,ρp
µ)≥1
2.
(5.6)
For
x∈Eα,ε1,ρ1
µ∩lim supn→+∞Bn
, consider
nx≥1
large enough so that
x∈Bnx
,
4rnx≤ρ1
, and
r−ε1
nx≥max ¶4Qd4α−ε1, ρ−d/τd
2©.
(5.7)
Set
Lx=B(x, 4rnx).
(5.8)
Doing so for every
x∈Eα,ε1,ρ1
µ∩lim supn→+∞Bn
provides us with a Besicovith
covering
F1=Lx:x∈Eα,ε1,ρ1
µ∩lim supn→+∞Bn
such that for every
x
, the
ball
Lx
is naturally associated with an integer
nx≥1
such that
x∈Bnx
and
|Lx|= 8rnx.
Also, the shrunk rectangle
Rnx
veries
Rnx⊂Bnx⊂Lx.
This is
illustrated by Figure 5.1.

86
CHAPTER 5: ANISOTROPIC UBIQUITY FOR QUASI-BERNOULLI
MEASURES
Figure 5.1: Representation of
Bnp
,
|Lp|= 8rnp
and
Rnp
.
Applying now Proposition 4.1.2 with
v= 1
, from the family
F1
one can
extract
Qd:= Qd,1
nite or countable families of balls
F1
i
,
1≤i≤Qd
, such
that:

∀1≤i≤Qd,∀L=L′∈ F1
i
, it holds that
L∩L′=∅,

Eα,ε1,ρ1
µ∩lim supn→+∞Bn⊂S1≤i≤QdSL∈F1
iL.
Since
µEα,ε1,ρ1
µ∩lim supn→+∞Bn≥1
2
, there exists
1≤i1≤Qd
such that
µ[
L∈Fi1
L≥µ(Eα,ε1,ρ1
µ∩lim supn→+∞Bn)
Qd≥1
2Qd
.
(5.9)
Denote by
(L(1)
k)k∈N
the sequence of balls such that
F1
i1=¶L(1)
k©k∈N
,
(x(1)
k)k∈N
the sequence of points such that for all
k∈N
,
L(1)
k=Lx(1)
k
, and set
r(1)
k=rx(1)
k
.
There exists
N1∈N
so that
µ[
1≤k≤N1
L(1)
k≥
µSL∈F1
i1
L
2.
Set
F1=¶L(1)
k©1≤k≤N1
. One has
µ[
L∈F1
L≥1
4Qd
.
(5.10)
Remember that with every ball
L(1)
k
are naturally associated the ball
Bn(1)
k
and
the rectangle
Rn(1)
k
, where
n(1)
k=nx(1)
k
; set
R(1)
k=Rn(1)
k
. Then dene
K1
, the

5.2 PROOF OF THEOREM 5.1.5 87
rst generation of the Cantor set by setting
K1=¶R(1)
k©1≤k≤N1
and
K1=[
R∈K1
R.
Finally, measure
η1
on the algebra generated by
K1
is obtained by concentrating
the
µ
-measure of the balls
Lx
on the rectangle
Rnx
. More precisely, for
1≤
k≤N1
set
η1(R(1)
k) = µÄL(1)
kä
P1≤k′≤N1µL(1)
k′.
Since for all
1≤k≤N1
, the center
x(1)
k
of
L(1)
k
belongs to
Eα,ε1,ρ1
µ
, recalling
that
|Lx(1)
k|/2 = 4rn(1)
k≤ρ1
, the disjointness of the
L(1)
j
, as well as the inequality
(5.10), we get that for all
1≤k≤N1
,
η1(R(1)
k)≤4Qd4rn(1)
kα−ε1≤rn(1)
kα−2ε1,
(5.11)
where (5.7) has been used.
Step 2: Constructing the second generation
This step consists of two sub-steps: First we associate a set of dyadic cubes
with each rectangle previously obtained, and then we work inside each of these
cubes.
Sub-step 2.1: A set of dyadic cubes inside each
R
of
K1
Consider a rectangle
R
. There exists an orthogonal matrix
O∈ Od(R)
, a point
x∈[0,1]d
and
0< ℓd≤ℓ2≤... ≤ℓ1
such that
R=x+Oe
R,
with
e
R=
d
Y
i=1
[0, ℓi].
Set
p=−ölog2Äℓd
8√däù
. Intuitively,
2−p≈ℓd
8√d
, so that there are some
cubes included in
R
with side-length
2−p.
We associate with
R
the set of dyadic
cubes (see Figure 5.2)
C(R) = (D∈ Dp:D⊂R, D =
d
Y
i=1
[ki2−p,(ki+ 1)2−p],8|ki,∀1≤i≤d).
Observe that
C(R)
consists in dyadic cubes of generation
p
inside
R
that
are quite far from each other. This will ensure that the rectangles used at a

88
CHAPTER 5: ANISOTROPIC UBIQUITY FOR QUASI-BERNOULLI
MEASURES
Figure 5.2: Representation of the lacunarity of
C(R)
inside
R
.
given generation of the construction of the Cantor set are well separated. Also,
there exist a constant
Cd≥1
depending only on the dimension
d
, such that
the side length
2−p
of each
C∈ C(R)
satises
C−1
dℓd≤2−p≤Cdℓd
, as well as
a constant
κd≥1
such that
κ−1
d
d
Y
i=1
ℓi
ℓd≤#C(R)≤κd
d
Y
i=1
ℓi
ℓd
.
Recalling (5.2), for every
n∈N
, one gets
κ−1
d·rPd
i=1 τi−τd
n≤#C(Rn)≤κd·rPd
i=1 τi−τd
n.
(5.12)
Now we construct a measure
η2
, which renes the measure
η1
by dis-
tributing the mass uniformly between the cubes of
C(R)
for
R∈ K1
. For every
1≤k≤N1
and every
D∈ C(R(1)
k)
, set
η2(D) = η1(R(1)
k)
#C(R(1)
k).
By construction,
η2(R(1)
k) = η1(R(1)
k).
Recalling (5.11) and (5.12), one gets
η2(D) = η1(R(1)
k)
#C(R(1)
k)≤rn(1)
kα−2ε1
κ−1
d·rn(1)
kPd
i=1 −τd+τi
=κd·rτd
n(1)
kα−2ε1+Pd
i=1 τd−τi
τd.
(5.13)

5.2 PROOF OF THEOREM 5.1.5 89
Sub-step 2.2: Construction in each cube of
C(R)
We start with preliminary observations about the measure
µ
. Recall Deni-
tion 5.1.1. Since
µ
is a quasi-Bernoulli measure, for every
q∈N
, every
D∈ Dq
such that
µ(D)>0
, for every
x∈[0,1]d
and
r > 0
such that
B(x, r)⊂D
, due
to (5.1) one has
µ(B(x, r)) = µT−1
D(TD(B(x, r)))=µ(D)µDBTD(x),r
2−q
≤Cµµ(D)µBTD(x),r
2−q.
Thus, for all
x∈[0,1]d
and
r > 0
such that
B(x, r)⊂[0,1]d
one has
µB(T−1
D(x), r2−q)≤Cµµ(D)µ(B(x, r)).
(5.14)
Also, for every
p∈N
, (5.1) yields
µD(Eα,εp,ρp
µ)≥µ(Eα,εp,ρp
µ)
Cµ≥1
2Cµ
.
(5.15)
Moreover,
T−1
D(Eα,εp,ρp
µ)
=T−1
D(x) : ∀r≤ρp, B(x, r)⊂[0,1]d, µ(B(x, r)) ≤rα−εp
=nT−1
D(x) : ∀r≤ρp, B(x, r)⊂[0,1]d, µ TDBT−1
D(x),r
2q≤rα−εpo,
and using (5.14), one gets
T−1
D(Eα,εq,ρq
µ)
⊂ßT−1
D(x) : ∀r≤ρp, B(x, r)⊂[0,1]d,µ(B(T−1
D(x), r2−q))
µ(D)≤Cµrα−εp™
=ßy∈D:∀r≤ρp2−q, B(y, r)⊂D, µ(B(y, r))
µ(D)≤Cµr
2−qα−εp™.
It follows that if we x
p
as above and set
Eεp
D= lim sup
n→+∞
Bn∩ßy∈D:∀r≤ρp2−q, B(y, r)⊂D, µ(B(y, r))
µ(D)≤Cµr
2−qα−εp™,
(5.16)
then by Denition 5.1.1 and the fact that
µ(lim supn→+∞Bn) = 1,
we have
µ(Eεp
D) = µ(T−1
D(TD(Eεp
D))) = µ(D)µD(TD(Eεp
D)) ≥µ(D)µ(Eα,εp,ρp
µ)
Cµ≥µ(D)
2Cµ
,
(5.17)

90
CHAPTER 5: ANISOTROPIC UBIQUITY FOR QUASI-BERNOULLI
MEASURES
where we used (5.15).
We now continue the construction. Consider
R∈ K1
. Fix
D∈ C(R)
.
Remember that
p(D)
is the unique integer such that
D∈ Dp(D)
. The set
Eε2
D
is well dened since
µ(D)>0
(the measure
µ
has been supposed to be fully
supported on
[0,1]d
). For every
x∈Eε2
D
, consider
nx
large enough so that:
•x∈Bnx
,
•nx≥2
and
4rnx≤ρ22−p(D),
and
r−ε2
nx≥max ¶4CµQd·η2(D)(4 ·2p(D))α−ε2, ρ−d/τd
3©.
(5.18)
Set
Lx=B(x, 4rnx)
, as in step 1 (see (5.8)). By repeating the same argument
as in step 1, one can extract from
{Lx:x∈Eε2
D}
a nite number
ND
of balls,
L(D)
1=Lx(D)
1, ..., L(D)
ND=Lx(D)
ND
such that for all
1≤k1=k2≤ND
one has
L(D)
k1∩L(D)
k2=∅
and by (5.17)
µ[
1≤k≤ND
L(D)
k≥µ(Eε2
D)
2≥µ(D)
4QdCµ
.
(5.19)
and with each ball
L(D)
k
are associated the ball
Bnx(D)
k
and the rectangle
Rnx(D)
k
,
that we denote by
B(D)
k
and
R(D)
k
respectively; we also set
r(D)
k=rnx(D)
k
. Then
dene the collection of rectangles of second generation by setting
K(R) = [
D∈C(R)¶R(D)
k©1≤k≤ND
and
K2=[
R∈K1K(R),
and
K2=[
R∈K2
R.
One extends further the measure
η1
to the algebra generated by the elements
of the set
K1SSR∈K1C(R)SK2
by distributing the mass according to
µ
at
that scale. More precisely, for all
R∈ K1
,
D∈ C(R)
and
1≤k≤ND
, one sets
η2(R(D)
k) = η2(D)µ(L(D)
k)
P1≤k′≤NDµ(L(D)
k′)
Note the following facts:

If
R∈ K1
,
D, D′∈ C(R)
,
1≤k≤ND
and
1≤k′≤ND′
are such that
R(D)
k=R(D′)
k′∈ K(R)
, then
3B(D)
k∩3B(D′)
k′=∅.

5.2 PROOF OF THEOREM 5.1.5 91

If
R∈ K1
,
D∈ C(R)
and
1≤k≤ND
, using the second assertion of
(5.18) and the fact that the ball
L(D)
k
is centered on
Eε2
D
, then
µ(L(D)
k)
µ(D)≤CµÇ4r(D)
k
2−p(D)åα−ε2
so that by (5.19) and the third assertion of (5.18), we get
η2(R(D)
k)≤η2(D)4QdCµ(4 ·2p(D))α−ε2·(r(D)
k)α−ε2≤(r(D)
k)α−2ε2.
(5.20)
Further steps: Induction scheme
We proceed as in step 2. Suppose that
p≥2
, and for all
1≤q≤p
, a set
Kq
and a measure
ηq
, dened on the algebra generated by the elements of
S1≤p≤qKpSR∈KpC(R)
, have been constructed in such a way that (5.11) holds
and:
(i)
For all
1≤q≤p
,
Kq
is a nite subset of
{Rn}n≥q.
(ii)
For all
2≤q≤p
, for all
R∈ Kq
, there exists
R′∈ Kq−1
and
D∈ C(R′)
such that
R⊂D
; one denotes by
¶R(D)
k©1≤k≤ND
the family of rectangles
of
Kq
included in
D
.
(iii)
For all
1≤q≤p−1
and
R∈ Kq
, if
rτd
is the length of the smallest side
of
R
, then
(rτd)−εq≥ρ−d
q+1.
(5.21)
(iv)
For all
2≤q≤p
,
R∈ Kq−1
,
D∈ C(R)
and
1≤k≤ND
, with
the rectangle
R(D)
k
are naturally associated a point
x(D)
k∈Eεq
D
, a ball
L(D)
k=BÄx(D)
k,4r(D)
kä
, as well as some integer
nk∈N
, such that
nk≥q
,
x(D)
k∈B(D)
k:= Bnk=B(xnk, rnk)
,
R(D)
k=Rnk
,
r(D)
k=rnk
and
4r(D)
k≤
2−p(D)ρq
. In particular, due to (5.16), one has
µ(L(D)
k)
µ(D)≤CµÇ4rn(D)
k
2−p(D)åα−εq
.
(5.22)
(v)
For all
2≤q≤p
,
R∈ Kq−1
,
D, D′∈ C(R)
,
1≤k≤ND
and
1≤k′≤ND′
such that
R(D)
k=R(D′)
k′
, one has
3B(D)
k∩3B(D′)
k′=∅.
(vi)
For all
1≤q≤q′≤p
and
R∈ Kq
,
ηq(R) = ηq′(R).

92
CHAPTER 5: ANISOTROPIC UBIQUITY FOR QUASI-BERNOULLI
MEASURES
(vii)
For all
2≤q≤p
,
R∈ Kq−1
,
D∈ C(R)
and
1≤k≤ND
, one has
ηq(D) = ηq−1(R)
#C(R)
and
(r(D)
k)−εq≥4CµQd·ηq(D)(4 ·2p(D))(α−εq).
(5.23)
(viii)
For all
2≤q≤p
,
R∈ Kq−1
,
D∈ C(R)
and
1≤k≤ND
, one has
X
1≤k′≤ND
µ(L(D)
k′)≥µ(D)
4Qd
(5.24)
and
ηq(R(D)
k) = ηq(D)·µ(L(D)
k)
P1≤k′≤NDµ(L(D)
k′).
(5.25)
Notice that by (5.22), (5.23) (5.24) and (5.25), for all
2≤q≤p
,
R∈ Kq−1
,
D∈ C(R)
and
1≤k≤ND
, one has
ηq(R(D)
k) = ηq(D)·µ(L(D)
k)
P1≤k′≤NDµ(L(D)
k′)≤ηq(D)µ(L(D)
k)
µ(D)(4Qd)−1
≤ηq(D)4r(D)
kα−εq4QdCµ2p(D)α−εq≤r(D)
kα−2εq.
(5.26)
Thus, for all
2≤q≤p
,
R∈ Kq−1
and
D∈ C(R)
, denoting by
rτd
the length of
the smallest side of
R
, by (5.11), (5.12), (5.23),(5.26) and
(vi)
, one has
ηq(D) = ηq−1(R)
#C(R)≤rα−2εq−1
κ−1
d·rPd
i=1 −τd+τi≤κdrα−2εq−1+Pd
i=1 τd−τi
≤κdrτdα−2εq−1+Pd
i=1 τd−τi
τd.
(5.27)
Let us now explain the induction. Take
R∈ Kp
and
D∈ C(R)
. For every
x∈Eεp+1
D
, consider an integer
nx
large enough so that:

x∈Bnx
,

nx≥p+ 1
,
4rnx≤ρp+12−p(D)
, and
r−εp+1
nx≥max Å4α−εp+1 ηp(R)
#C(R)4QdCµ2p(D)(α−εp+1), ρ−d/τd
p+2 ã.
Using Proposition 4.1.2 with
v= 1
, one can extract from the covering of
Eεp+1
D
,
Lx:= B(x, 4rnx) : x∈Eεp+1
D
, a nite set of balls
F(D) := ¶L(D)
k:= L(D)
xk©1≤k≤ND
such that

5.2 PROOF OF THEOREM 5.1.5 93

∀k=k′≤ND
,
L(D)
k∩L(D)
k′=∅.
In particular,
3Bnx(D)
k∩3Bnx(D)
k′
=∅,

writing
Qd=Qd,1,
one has
µ[
1≤k≤ND
L(D)
k≥1
2µ(Eεp+1
D)≥µ(D)
4QdCµ
.
(5.28)
Consider the collection of rectangles naturally associated with the balls
L(D)
k
Kp+1(R) = [
D∈C(R)ßR(D)
k:= Rnx(D)
k™1≤k≤ND
.
Then dene
Kp+1 =[
R∈KpK(R)
and
Kp+1 =[
R∈Kp+1
R.
The probability measure
ηp
can be extended from the algebra generated
by the elements of
S1≤p≤pKqSSR∈KpC(R)
to the algebra generated by the sets
of the union
S1≤q≤p+1 KqSSR∈KpC(R)
as follows: For
R∈ Kp
and
D∈ C(R)
,
we impose that
ηp+1(R) = ηp(R)
and
ηp+1(D) = ηp(R)
#C(R).
(5.29)
And then, for
R∈ Kp
,
D∈ C(R)
and
1≤k≤ND
, we set
ηp+1(R(D)
k) = ηp+1(D)·µ(L(D)
k)
P1≤k′≤NDµ(L(D)
k′).
(5.30)
It is easily checked that properties
(i)
to
(viii)
hold for
p+ 1
and this ends the
induction.
Last step: the Cantor set and some of its properties.
Set
K0= [0,1]d
and
η0([0,1]d) = 1.
Dene
K=[
p∈NKp
and
K=\
p∈N
Kp.
By construction, item
(i)
of the recursion implies that
K⊂lim supn→∞ Rn
.
Now, for each
p≥1
, let
eηp
be the element of
P([0,1]d)
supported on
Kp
and
such that for every
R∈ Kp
the restriction of
ηp
to
R
has
ηp(R)
Ld(R)
as density with
respect to
Ld
|R
. It is easily seen, due to the separation property of the elements

94
CHAPTER 5: ANISOTROPIC UBIQUITY FOR QUASI-BERNOULLI
MEASURES
of
Kp
, for all
p∈N
, that
(eηp)p∈N∗
converges weakly to a Borel probability
measure
η
such that
η(R) = ηp(R)
for all
p∈N
and
R∈ Kp
.
Note that by construction the following properties hold:

Uniform separation property:
For all
p∈N
and
n∈N
such that
Rn∈
Kp
, if
n1, n2∈N
such that
Rn1=Rn2∈ K(Rn) = {R′∈ Kp+1 :R′⊂Rn}
,
one has
3Bn1∩3Bn2=∅
. In the case where
Rn1
and
Rn2
are elements of
the same
D∈ C(Rn)
, this follows from
(v)
; otherwise, this follows from
the fact that two distinct elements
D
and
D′
of
C(Rn)
are distant from
each other by at least
8·2−p(D)
, where, as before,
p(D)
is the unique
integer such that
D∈ Dp(D)
.

The estimates (5.26) and (5.27) show (by induction) that for all
p∈N∗
and
n∈N
such that
Rn∈ Kp
one has
η(Rn)≤rα−2εp
n,
and for all
D∈ C(Rn)
,
η(D)≤κdrα−2εp+P1≤i≤dτd−τi
n=κdrτd
nα−2εp+P1≤i≤dτd−τi
τd.
(5.31)
Upper bound for the
η
-measure of a ball.
Let
C
be a ball (recall that it is an Euclidean cube) of side length
r
contained
in
[0,1]d
. Several cases are distinguished.
•
When
C
intersects
Kp
for at most nitely many
p∈N
, it is clear that
η(C)=0
, and we set
pC= +∞
.
•
When
C
intersects a unique rectangle of
Kp
, say
RnC(p)
, for innitely
many
p∈N
, then
η(C)≤η(RnC(p))≤rα−2εp
nC(p)
for innitely many
p
, so
η(C)=0
.
Again, we set
pC= +∞
.
•
Suppose now that we are not in one of the previous cases. There exists
pC∈N
such that if
p≤pC
,
C
intersects a unique rectangle of
Kp
and if
p≥pC+ 1
,
C
intersects at least two rectangles of
Kp
. Denote by
RnC
the
unique rectangle in
KpC
intersecting
C
. Let
v > 0
be such that
r=rv
nC
. Again,
several cases are distinguished.
(i)
Suppose
r≥rτd
nC
(i.e.
v≤τd
): Suppose, moreover, that
r < rnC
, i.e.

5.2 PROOF OF THEOREM 5.1.5 95
1< v ≤τd
. Remember that for
D∈ C(RnC)
one has (see (5.31))
η(D)≤κdrnCα−2εpC+P1≤i≤dτd−τi.
Also, there exists
eκd>0
, depending on
d
so that
#{D∈ C(RnC) : D∩C=∅} ≤ eκdY
i:τi<v rv
nC
rτd
nCY
i:τi≥vrτi
nC
rτd
nC
≤eκdr−dτd+Pi:τi<v v+Pi:τi≥vτi
nC.
Provided that
κd
was chosen larger than
eκd
at rst, one gets
This gives the following upper bound for
η(C)
:
η(C)≤X
D∈C(RnC):D∩C=∅
η(D)
≤(# {D∈ C(RnC) : D∩C=∅})·κdrnCα−2εpC+P1≤i≤dτd−τj
≤κ2
drnC−2εpCrnC−dτd+α+Pi:τi<v v+Pi:τi≥vτi+P1≤i≤dτd−τi
≤κ2
drnC−2εpCrnCα+Pi:τi<v v−τi
≤κ2
dr−2εpCr
α+Pi:τi<v v−τi
v.
The mapping
f:v7→ α+Pi:τi<v v−τi
v
reaches its minimum at one of the
τi
, say
τi0
with
1≤i0≤d
. This can be rephrased as
s(µ, τ) = min1≤i≤d(α+P1≤j≤iτi−τj
τi) =
f(τi0)
. It follows that
η(C)≤κ2
drs(µ,τ)−2εpC.
(5.32)
On the other hand, if
r≥rnC
, i.e.
v≤1
, then by (5.26), one has
η(C)≤η(RnC)≤rα−2εpC
nC≤rα−2εpC,
and (5.32) holds as well, since
α=f(τ1)≥s(µ, τ)
.
(ii)
Suppose now that
r < rτd
nC
(i.e.
v > τd
):
Recall that
rτd
nC
is the length of the smallest side of the rectangle
RnC
.
Since
C
has side length less than
rτd
nC
, and the side length of the cubes of
C(RnC)
is larger than or equal to
C−1
drτd
nC
, one deduces that
C
intersects at
most
e
Cd
of those cubes, where
e
Cd
depends on
d
only. For all
D∈ C(RnC)
, such
that
C∩D=∅
, denote by
R(D)
k1, ..., R(D)
kNC,D
the rectangles included in
D
that
intersect
C
.
•
Suppose rst that
20r≤2−p(D)ρpC+1
(where
D∈ Dp(D)
): Note that

96
CHAPTER 5: ANISOTROPIC UBIQUITY FOR QUASI-BERNOULLI
MEASURES
for all
1≤i=j≤NC,D
,
3B(D)
ki∩3B(D)
kj=∅
. Also,
C
intersects both
B(D)
ki
and
B(D)
kj
, and by construction, since
L(D)
ki∩L(D)
kj=∅
and
|L(D)
kj|= 4|B(D)
kj|
, we
have
r≥r(D)
kj
. By Lemma 5.2.1 applied to each pair
{A=C, B =B(D)
kj}
and
q= 3
, one gets
S1≤i≤NC,D 3B(D)
ki⊂5C
. In particular,
S1≤i≤NC,D L(D)
ki⊂10C
since
L(D)
ki⊂5B(D)
ki
for each
i
. Consequently,
X
1≤i≤NC,D
µ(L(D)
ki)≤µ(10C).
Further recall that, by item
(iv)
of the recurrence scheme, for any
1≤i≤NC,D
the ball
L(D)
ki
is centered on
EεpC+1
D
. Thus there is
x∈EεpC+1
D∩10C
. Since one
has
10C⊂B(x, 20r)
and
20r
2−p(D)≤ρpC+1
, by (5.22) we get
µ(10C)≤µ(B(x, 20r)) ≤Cµµ(D)20r
2−p(D)α−εpC+1 .
(5.33)
It follows from (5.28), (5.30) and (5.33) that
η(C∩D)≤X
1≤i≤NC,D
η(R(D)
ki)≤η(D)X
1≤i≤NC,D
µ(L(D)
ki)
P1≤j≤NDµ(L(D)
j)
≤η(D)X
1≤i≤NC,D
µ(L(D)
ki)
(4Qd)−1µ(D)
≤4Qd
η(D)
µ(D)µ(10C)
≤Cµη(D) 4Qd20r
2−p(D)α−εpC+1 .
This yields
η(C)≤X
D∈C(RnC):C∩D=∅
η(C∩D)
≤e
CdCµmax
D∈C(RnC):C∩D=∅η(D) 4Qd20r
2−p(D)α−εpC+1 .
Moreover by (5.27), for each
D∈ C(RnC)
such that
C∩D=∅
,
η(D)≤κd22p(D)εpC2−s(µ,τ)p(D),
hence
η(C)≤e
CdCµκd4Qd22p(D)εpC2−p(D)s(µ,τ)20r
2−p(D)α−εpC+1 .
Since
Cd2−p(D)≥rτd
nC≥r
and the sequence
(εp)p≥1
is decreasing and bounded,
it follows that for some constant
γ
depending only on the dimension
d
and
µ
,

5.2 PROOF OF THEOREM 5.1.5 97
one has
η(C)≤γ r−3εpCrα
2−p(D)(α−s(µ,τ)) =γ r−3εpCr
2−p(D)α−s(µ,τ)rs(µ,τ).
Thus, as
Cd2−p(D)≥r
and
s(µ, τ)≤α
(so that
t > 07→ tα−s(µ,τ)
is non
decreasing), we nally obtain
η(C)≤γ Cα−s(µ,τ)
drs(µ,τ)−3εpC≤γ Cα
drs(µ,τ)−3εpC.
•
Suppose now that
20ρpC+1 2−p(D)≤r < rτd
nC
: Again, by denition
of
p(D)
, one has
rτd
nC≤Cd2−p(D)
. Consequently,
C
is covered by at most
⌊(Cd/20ρpC+1) + 1⌋d
cubes of side length
20ρpC+1 2−p(D)
. Denoting these cubes
by
D1, . . . , Dk
, and recalling (5.21), the previous estimate yields
η(C)≤
k
X
i=1
η(Di)≤ ⌊(Cd/20ρpC+1)+1⌋dγ Cα
d20ρpC+1 2−p(D)s(µ,τ)−3εpC
≤γ1ρ−d
pC+1 rs(µ,τ)−3εpC≤γ1rs(µ,τ)−4εpC
for some constant
γ1
depending only on
d
and
µ
(we used that
20ρpC+1 2−p(D)≤
r
to get the third inequality, and
(iii)
as well as the inequality
εpC≥εpC+1
to
get the fourth one).
To conclude the proof, note that due to the uniform separation property
outlined after the last step of the construction of
(K, η)
,
p(r) = inf{pC:C
is ball of radius r included in
[0,1]d}
tends to
+∞
as
r
tends to
0
.
Combining the previous estimates, setting
eγ1= max {γ1, γ.Cα
d, κ2
d}
, we
nally get
η(C)≤eγ1rs(µ,τ)−4εp(r).
In particular, for any
p∈N
, setting
rp=1
2sup {r:p(r)≤p}
, it holds that for
any
r≤rp
, any
C
of radius
r
,
η(C)≤eγ1rs(µ,τ)−4εp.
By Lemma 4.2.4, since
η(K) = 1
, it holds that
dimH(K)≥s(µ, τ)−4εp.

98
CHAPTER 5: ANISOTROPIC UBIQUITY FOR QUASI-BERNOULLI
MEASURES
Letting
p→+∞
proves Theorem 5.1.5.
The next section introduces some ley notion in order to establish a more
general mass transference principle than the one we just established.

Chapter 6
Asymptotically covering sequences
of balls and essential content
This chapter comes as a preamble of Chapter 7. First, in Section 6.1 given
a sequence of balls
(Bn)n∈N,
one investigates the link between satisfying the
condition
µ(lim supn→+∞Bn)=1
and verifying a certain covering property, we
call
µ
asymptotically covering ( Denition 6.1.1). This property is usually used
in mass transference principles when the reference measure is doubling on its
support ([11]).
In a second time, in Section 6.2, one denes and study the key geometric
notion we will need to establish our main ubiquity theorem, Theorem 7.1.2,
which is the essential Hausdor content. Some explicit estimates of this geo-
metric quantity are given and some basic properties are established.
The last section, Section 6.1.3 gives a version of the Borel-Cantelli Lemma
suitable for limsup sets of balls.
6.1 Limsup sets of full measure and asymptoti-
cally covering sequences of balls
Let
(Bn)n∈N
be a sequence of balls of
[0,1]d
satisfying
|Bn| → 0.
An important
covering property, highlighted in [11] and used to establish ubiquity theorems
or mass transference principles in the case of the Lebesgue measure, is that for
any open set
Ω
and any
g∈N
, assuming that
Ld(lim supn→+∞Bn) = 1
it is
possible to nd disjoints balls
Bn1, ..., Bnk⊂Ω
with
ni≥g
for all
i
and
Ld[
1≤i≤k
Bni≥1
2Ld(Ω).

100 CHAPTER 6:
µ
-A.C SEQUENCES OF BALLS
In this section, one investigates, given any probability measure
µ
, the rela-
tionship between being verifying
µlim supn→+∞Bn= 1
and satisfying the
covering property stated above where
Ld
is replaced by
µ.
Denition 6.1.1.
Let
µ∈ M(Rd)
. The sequence
B= (Bn)n∈N
of balls of
Rd
is said to be
µ
-asymptotically covering (in short,
µ
-a.c) when there exists a
constant
C > 0
such that for every open set
Ω⊂Rd
and
g∈N
, there is an
integer
NΩ∈N
as well as
g≤n1≤... ≤nNΩ
such that:

∀1≤i≤NΩ
,
Bni⊂Ω,

∀1≤i=j≤NΩ
,
Bni∩Bnj=∅,

one has
µ [
1≤i≤NΩ
Bni!≥Cµ(Ω).
(6.1)
In other words, for any open set
Ω
and any
g > 0
, there exists a nite set
of disjoint balls of
{Bn}n≥g
covering a large part of
Ω
from the
µ
-standpoint.
This notion of
µ
-asymptotically covering is related to the way the balls of
B
are distributed according to the measure
µ
. In particular, given a measure
µ
, this property turns out to be slightly stronger than being of
lim sup
of full
µ
-measure, as illustrated by the following Theorem.
Theorem 6.1.2.
Let
µ∈ M(Rd)
and
B= (Bn:= B(xn, rn))n∈N
be a sequence
of balls of
Rd
with
limn→+∞rn= 0
.
1. If
B
is
µ
-a.c, then
µ(lim supn→+∞Bn)=1.
2. If there exists
v < 1
such that
µlim supn→+∞(vBn)= 1
, then
B
is
µ
-a.c.
Moreover, it results from the proof of the KGB-Lemma [11] that if the
µ
is doubling,
µlim supn→+∞Bn= 1 ⇔(Bn)n∈N
is
µ
-a.c.
One emphasizes also that, in general, the contracted sets we consider in
ubiquity theorems are much smaller than
vBn
for any
v < 1
. In particular, as
it will be highlighted later in this manuscript, this condition is not restrictive
at all for the cases we consider.
6.1.1 Consequences of the
µ
-asymptotic covering prop-
erty
One rst shows that the constant
C
in Denition 6.1.1 can be replaced by 1
if innite subsequences of balls are authorized. In fact Denition 6.1.1 ensures

6.1 LIMSUP AND
µ
-A.C PROPERTY 101
that any open set can be covered (with respect to the
µ
-measure) by disjoint
balls
Bn
of arbitrary large indices.
Lemma 6.1.3.
Let
µ∈ M(Rd)
and
B= (Bn:= B(xn, rn))n∈N
be a
µ
-a.c
sequence of balls of
Rd
with
limn→+∞rn= 0
.
Then for every open set
Ω
and every integer
g∈N
, there exists a subse-
quence
(B(Ω)
(n))⊂ {Bn}n≥g
such that:
1.
∀n∈N
,
B(Ω)
(n)⊂Ω,
2.
∀1≤n1=n2
,
B(Ω)
(n1)∩B(Ω)
(n2)=∅
,
3.
µÄSn≥1B(Ω)
(n)ä=µ(Ω).
In addition, there exists an integer
NΩ
such that for the balls
(B(Ω)
(n))n=1,...,NΩ
,
the conditions (1) and (2) are realized, and (3) is replaced by
µÄSNΩ
n=1 B(Ω)
(n)ä≥
3
4µ(Ω).
The last part of Lemma 6.1.3 simply follows from item (3) and the
σ
-
additivity of
µ
.
Proof.
The idea consists in covering
Ω
by pairwise disjoint balls amongst those
balls of
B
, such that their union has measure at least
Cµ(Ω)
, then in covering
the complementary of the union of those balls in
Ω
(that is still open) with at
least a proportion
C
of its measure, and so on.
More precisely, this is achieved as follows:
•
Step 1:
By application of Denition 6.1.1 to
Ω0:= Ω
and
g∈N
, there
exists
C > 0
and some integers
g≤n1≤... ≤nN0
so that the family of balls
F0:= ¶Bni:= B(0)
i©1≤i≤N0
is pairwise disjoint and
µ(S1≤i≤N0Bni)≥Cµ(Ω).
•
Step 2:
Setting
Ω1= Ω \SL∈F0L
, applying Denition 6.1.1 to
Ω1
with the integer
g
provides us with a family
F1
of pairwise disjoint balls
B(1)
1, ..., B(1)
N1∈ {Bn}n≥g
such that
∀1≤i≤N1B(1)
i⊂Ω1
and
µ([
1≤i≤N2
B(1)
i)≥Cµ(Ω1).

102 CHAPTER 6:
µ
-A.C SEQUENCES OF BALLS
One sets
F1=F0SF1
. One sees that
µ [
L∈F1
L!=µ [
L∈F0
L!+µ [
L∈G1
L!
≥µ [
L∈F0
L!+C µ(Ω) −µ [
L∈F0
L!!
≥(1 −C)µ [
L∈F0
L!+Cµ (Ω)
≥(C+C(1 −C))µ(Ω).
Observe that the balls of
F0
and
F1
are disjoint by construction.
•
Following steps :
Proceeding recursively, and applying the exact same
argument as above, one constructs an increasing sequence of families
(Fi)i∈N
and a decreasing sequence of open sets
Ωi
such that:

∀i∈N
,
L∈ {Bn}n≥g
and
∀L∈ Fi
,
L⊂Ωi⊂Ω,

∀i∈N
,
∀L=L′∈ Fi
,
L∩L′=∅
,

∀i=j∈N
,
∀L∈ Fi
and
∀L∈ Fj
,
L∩L′=∅
,

∀i∈N
,
µSL∈FiL≥µ(Ω) P1≤k≤iC(1 −C)k−1.
Finally, setting
F=Si∈NFi
, one sees that
F
is constituted by pairwise
disjoint balls chosen amongst
{Bn}n≥g
satisfying
µ(Ω) ≥µ [
L∈F
L!≥µ(Ω) X
k≥1
C(1 −C)k−1=µ(Ω),
(6.2)
so that
F
fullls the conditions of Lemma 6.1.3.
An easy consequence is the following.
Corollary 6.1.4.
Let
µ∈ M(Rd)
and
(Bn)n∈N
be a
µ
-a.c sequence of balls.
Then for any Borel set
E
, for any
g∈N
, there exists a sub-sequence of balls
(B(E)
(n))⊂ {Bn}n≥g
such that:
1.
∀1≤n1=n2
,
B(E)
(n1)∩B(E)
(n2)=∅,
2.
µSn∈NB(E)
(n)∩E=µ(E),
3.
µSn∈NB(E)
(n)≤µ(E) + ε,

6.1 LIMSUP AND
µ
-A.C PROPERTY 103
Proof.
By outer regularity, there exists an open set
Ω
such that
E⊂Ω
and
m(Ω) ≤µ(E) + ε.
Applying Lemma 6.1.3 to
Ω
, the sequence
(Bn)n∈N
fullls
the condition of Corollary 6.1.4.
6.1.2 Proof of Theorem 6.1.2
(1)
Assume rst that
B= (Bn)n∈N
is
µ
-a.c, and let us prove that
µ(lim supn→+∞Bn) =
1
.
For every
g∈N
, applying Lemma 6.1.3, there exists a sub-family of balls,
Fg⊂ {Bn}n≥g
such that
µ(SL∈FgL) = µ(Rd)=1.
In particular,
µ(Sn≥gBn) =
1
for every
g≥1
, and
µ(lim supn→+∞Bn) = µ(Tg≥1Sn≥gBn)=1
.
(2)
Suppose next that there exists
v < 1
such that
µ(lim supn→+∞vBn) =
1
, and let us show that
B
is
µ
-a.c.
Let
Ω
be an open set in
Rd
. Our goal is to nd a constant
C
such that
the conditions of Denition 6.1.1 are realized.
Let
E= Ω ∩lim supn→+∞vBn
. For every
y∈E
, consider an integer
ny≥g
large enough so that
y∈vBny
and
B(y, 2rny)⊂Ω.
This is possible
since
limn→+∞rn= 0
.
Since
y∈vBny
, one has
B(y, (1 −v)rny)⊂Bny⊂B(y, (1 + v)rny)⊂B(y, 2rny),
(6.3)
and the family
F=B(y, (1 −v)rny) : y∈E
covers
E
by balls centered on
E
.
Applying Proposition 4.1.2 with constant
v′=1−v
2<1
allows to extract
from
F
nite or countable sub-families
F1, ..., FQd,v′
such that:

∀1≤i≤Qd,v′
,
L=L′∈ Fi
, one has
1
v′L∩1
v′L′=∅.

E
is covered by the families
Fi
, i.e. (4.2) holds true.
Now,
µ(Ω) = µ(E)≤µÄSQd,v′
i=1 SL∈FiLä
. There must exist
1≤i0≤Qd,v′
such that
µÑ[
L∈Fi0
Lé≥1
Qd,v′
µ(E) = 1
Qd,v′
µ(Ω).
There exist
L1
,
L2
, ...
LN
balls of
Fi0
such that
µ [
1≤k≤N
Lk!≥1
2Qd,v′
µ(Ω),

104 CHAPTER 6:
µ
-A.C SEQUENCES OF BALLS
Notice the following facts:

∀1≤i≤Qd,v′
, every
L∈ Fi
is naturally associated with some
y∈E
and
some ball
Bny
, with
L⊂Bny⊂Ω
,

∀1≤i≤Qd,v′
, if
L∈ Fi
is associated with
y∈E
and
Bny
and
L′∈ Fi
is associated with
y′∈E
and
Bny′
, then
1
v′L∩1
v′L′=∅
implies by (6.3)
that
Bny∩Bny′=∅
.
The rst fact implies that there exist
N
integers
n1
, ...,
nN
such that
Bnk⊂Ω
and
µ [
1≤k≤N
Bnk!≥1
2Qd,v′
µ(Ω),
The second fact implies that these balls
Bnk
,
k= 1, ..., N
are pairwise disjoint.
This exactly proves that
B
is
µ
-a.c.
6.1.3 A version of Borel-Cantelli Lemma
In this manuscript, one mainly focuses on establishing Hausdor dimension of
limsup sets knowing that a certain limsup set of balls has full measure. In many
situations, proving that those limsup sets have full measure is straightforward.
Given
µ∈ M(Rd)
, when establishing that a sequence of balls
(Bn)n∈N
veries
µ(lim sup
n∈N
Bn) = 1
is not straightforward, it is convenient to have a tool at our disposal to be able
to determine whether or not it is the case. The case where the measure involved
is doubling is treated by Beresnevich and Velani.
Theorem 6.1.5
([12])
.
Let
µ∈ M(Rd)
be a doubling measure and
(Bn)n∈N
a
sequence of balls centered in
supp(µ)
such that
|Bn| → 0.
Then the two following
assertions are equivalent:
1.
µ(lim supn∈NBn) = 1.
2. There exists
C > 1
such that for any open ball
B
centered on
supp(µ),
there exists a sub-sequence
(LB,n)n∈N
of
(Bn)n∈N
satisfying:

for every
n∈N, LB,n ⊂B,

P+∞
n=0 µ(LB,n) = +∞,

6.1 LIMSUP AND
µ
-A.C PROPERTY 105

for innitely many
Q∈N,
Q
X
s,t=1
µ(LB,s ∩LB,t)≤C
µ(B) Q
X
n=1
µ(LB,n)!2
.
(6.4)
Thanks to Theorem 6.1.2, one can provide a version of this theorem valid
for any probability measure on
Rd
.
Proposition 6.1.6
(D. [24])
.
Let
(Bn)n∈N
be a sequence of closed balls satis-
fying
|Bn| → 0
and
µ∈ M(Rd)
be a probability measure.
(A): Assume that
(Bn)n∈N
is
µ
-a.c. Then there exists
C > 1
such that for any
open ball
B
, there exists a sub-sequence of
(Bn)n∈N
,
(Ln,B)n∈N
satisfying,
for any
n∈N
,
LB,n ⊂B
,
X
n≥0
µ(LB,n) = +∞
(6.5)
and for innitely many Q,
Q
X
s,t=1
µ(LB,s ∩LB,t)≤C
µ(B) Q
X
n=1
µ(LB,n)!2
.
(6.6)
(B): Assume that there exists
C > 1
such that for any open ball
B
, there exists
a sub-sequence of
(Bn)n∈N(Ln,B)n∈N
with, for any
n∈N, Ln,B ⊂B
,
satisfying
(6.5)
and
(6.6)
. Then
µ(lim supn→+∞Bn) = 1,
so that for any
κ > 1,(κBn)n∈N
is
µ
-a.c.
Proof.
Item
A
is proved in [12] (this part of the proof does not use the doubling
property of the measure). Moreover, it is also proved in [12] that, if there exists
C > 0
such that for any open ball
B
, there exists a sub-sequence of
(Bn)n∈N
(Ln,B)n∈N
with, for any
n∈N, Ln,B ⊂B
, satisfying (6.5) and (6.6), then
µ(lim sup
n→+∞
Bn∩B)≥1
Cµ(B).
The following lemma combined with Theorem 6.1.2 nishes the proof of Propo-
sition 6.1.6.
Lemma 6.1.7.
Let
E⊂Rd
. Assume that there exists
0< c < 1
such that for
any open ball
B
,
µ(E∩B)≥cµ(B).
Then
µ(E) = 1.
Proof.
Assume that
µ(E)<1
and set
A=Rd\E.
By hypothesis,
µ(A)>0.

106 CHAPTER 6:
µ
-A.C SEQUENCES OF BALLS
By Lemma 4.1.5, there exists an open ball
B
such that
µ(B)>0
and
µ(B∩A)≥(1 −c
2)µ(B).
This yields
µ(E∩B∩A) = µ(E∩B) + µ(A∩B)−µ((E∩B)∪(A∩B))
≥(c+ 1 −c
2−1)µ(B) = c
2µ(B)>0,
which implies
µ(E∩A)>0
, which is a contradiction.
Taking
c=1
C
and applying Lemma 6.1.7 nishes the proof of Proposition
6.1.6.
Remark 6.1.8.
A version of Proposition 6.1.6 might also be useful in more
general metric spaces. The only geometric property we used to prove Proposition
6.1.6 is actually Proposition 4.1.2 (which also implies Lemma 4.1.5), so that
Proposition 6.1.6 actually holds in any direction-limited spaces as dened in
[34].
6.2 Essential Hausdor content
In this section, we introduce the key geometric notion on which our main ubiq-
uity theorem, Theorem 7.1.2, relies.
Denition 6.2.1.
Let
µ∈ M(Rd)
, and
s≥0
. The
s
-dimensional
µ
-essential
Hausdor content at scale
t∈(0,+∞]
of a set
A⊂ B(Rd)
is dened as
Hµ,s
t(A) = inf {Hs
t(E) : E⊂A, µ(E) = µ(A)}.
(6.7)
One will almost exclusively look at these contents at scale
t= +∞
and
one refers to
Hµ,s
∞(A)
as the
s
-dimensional
µ
-essential Hausdor content of
A
. Basic properties of those quantities are studied in Sub-section 7.2.2, and
precise estimates of
Hµ,s
∞(A)
are achieved for the Lebesgue measure and weakly
conformal measures (so that those estimates holds for self-similar measures and
self-conformal measures) in this chapter.
Note that in [51, Theorem 3.1] the underlying geometric notion key to
handle the variety of shapes of the sets
(Un)n∈N
is the Hausdor content. It is
easily seen from (4.7) that the Hausdor content also carries some high scale
geometric information (because there is no restriction concerning the diameter
of the balls
(Bn)
in (4.7)). This will also be the case in this chapter to handle

6.2 ESSENTIAL HAUSDORFF CONTENT 107
not only the shape of the sets
(Un)n∈N
but also the geometric behavior related
to the measure
µ
at high scale in the sets
(Un)n∈N.
Also we work in this manuscript mainly with the
|| · ||∞
norm for conve-
nience. Any other norm could have been chosen, the corresponding quantities
would have been equivalent.
In (4.7), only closed balls are considered. Choosing open balls does not
change the value of (6.7) in Denition 6.2.1.
The following propositions are directly derived from the properties of the
standard Hausdor measures.
Proposition 6.2.2.
Let
µ∈ M(Rd)
,
s≥0
and
A⊂Rd
be a Borel set. The
s
-dimensional
Hµ,s
∞(·)
outer measure satises the following properties:
1. If
|A| ≤ 1
, the mapping
s≥07→ Hµ,s
∞(A)
is decreasing from
Hµ,0
∞(A)=1
to
limt→+∞Hµ,t
∞(A)=0
.
2.
0≤ Hµ,s
∞(A)≤min {|A|s,Hs
∞(A)}
.
3. For every subset
B⊂A
with
µ(A) = µ(B)
,
Hµ,s
∞(A) = Hµ,s
∞(B).
4. For every
δ≥1
,
Hµ, s
δ
∞(A)≥(Hµ,s
∞(A))1
δ.
5. For every
s > dimH(µ)
,
Hµ,s
∞(A) = 0.
Proof.
Items
(1)
,
(2)
,
(3)
directly follow from the denition. Item
(4)
is ob-
tained by concavity of the mapping
x7→ |x|1/δ
.
(5) By Denition 4.2.12, for any
s > dimH(µ)
, there exists a set
E
with
dimH(E)< s
and
µ(E)=1.
Using item (2), one has then
0≤ Hµ,s
∞(A) =
Hµ,s
∞(A∩E)≤ Hs
∞(A∩E)≤ Hs(E) = 0
.
6.2.1 Computation of essential content for the Lebesgue
measure
When the measure
µ
is the Lebesgue measure, the computations are quite easy.
Proposition 6.2.3.
Let
B=B(x, r)
be a ball in
Rd
, and
Ld
be the
d−
dimensional
Lebesgue measure. Then for any
0≤s≤d
,
HLd,s
∞(B) = HLd,s
∞(˚
B) = rs.

108 CHAPTER 6:
µ
-A.C SEQUENCES OF BALLS
Proof.
One starts rst by computing
HLd,d
∞(B)
.
Let
ε > 0
, and let
E⊂B
be a Borel set with
Ld(E) = Ld(B)
. Notice
rst that since
B
covers
E
, recalling that
Rd
is endowed with
|| · ||∞
one has
HLd,d
∞(E)≤ Hd
∞(B)≤ |B|d.
Consider a sequence of balls
(Ln)n∈N
such that
Hd
∞(E)≤X
n≥0|Ln|d≤(1 + ε)Hd
∞(E).
This implies
(1 + ε)|B|d≥(1 + ε)Hd
∞(B)≥(1 + ε)Hd
∞(E)≥X
n≥0|Ln|d
≥X
n≥0Ld(Ln)≥ Ld(E) = Ld(B) = |B|d.
Taking the inmum on the Borel sets
E⊂B
such that
Ld(E) = Ld(A)
gives
|B|d≤(1 + ε)HLd,d
∞(B).
In particular,
1
1 + ε|B|d≤ HLd,d
∞(B)≤ |B|d.
Letting
ε→0
shows that
HLd,d
∞(B) = |B|d
. This implies, with item
(4)
of
Proposition 6.2.2, that for any
δ≥1
,
|B|d
δ≥ HLd,d
δ
∞(B)≥(HLd,d
∞(B))1
δ=|B|d
δ,
hence the result.
6.2.2 Essential content for weakly conformal measures
Estimates of essential contents for weakly conformal measures are now estab-
lished.
Theorem 6.2.4
(D. [23])
.
Let
S
be a
C1
weakly conformal IFS of
Rd
.
Let
K
be the attractor of
S
and
µ
be a weakly conformal measure associ-
ated with
S
. Then,
For any
0≤s < dim(µ)
, for any
0< ε ≤1
2
, there exists a constant
c=c(d, µ, s, ε)>0
depending on the dimension
d
,
µ
,
s
and
ε
only, such that

6.2 ESSENTIAL HAUSDORFF CONTENT 109
for any ball
B=B(x, r)
centered on
K
and
r≤1
, for any open set
Ω
, one has
c(d, µ, s, ε)|B|s+ε≤ Hµ,s
∞(˚
B)≤ Hµ,s
∞(B)≤ |B|s
c(d, µ, s, ε)Hs+ε
∞(Ω ∩K)≤ Hµ,s
∞(Ω) ≤ Hs
∞(Ω ∩K).
(6.8)
For any
s > dim(µ)
,
Hµ,s
∞(Ω) = 0.
Remark 6.2.5.

The system
S
is not assumed to verify any separation condition.

When the maps are similarities, one still has, for any
s > dim(µ)
,
Hµ,s
∞(Ω) = 0
but for
s < dim(µ)
, there exists a constant
c(d, µ, s)
such that the following more
precise estimates holds true [25]:
c(d, µ, s)|B|s≤ Hµ,s
∞(˚
B)≤ Hµ,s
∞(B)≤ |B|s
and
c(d, µ, s)Hs
∞(Ω ∩K)≤ Hµ,s
∞(Ω) ≤ Hs
∞(Ω ∩K).
(6.9)
Proof.
Let us rst prove the above estimates for balls.
Proposition 6.2.6.
Let
µ
be a weakly conformal measure as in Denition
4.3.1. For any
0< ε ≤dim(µ)
, any
0≤ε′≤1
2
such that
dim(µ)−ε+ε′>0
,
there exists a constant
χ(d, µ, ε, ε′)>0
such that for any ball
B=B(x, r)
with
x∈K
(the attractor of the underlying IFS) and
r≤1
, one has
χ(d, µ, ε, ε′)|B|dim(µ)−ε+ε′≤ Hµ,dim(µ)−ε
∞(˚
B)≤ Hµ,dim(µ)−ε
∞(B)≤ |B|dim(µ)−ε.
In addition, for any
s > dim(µ)
,
Hµ,s
∞(B) = 0.
Proof.
Note rst that item (5) of Proposition 6.2.2 implies that for any
s >
dim(µ)
,
Hµ,s
∞(B) = 0.
Let us consider
0≤s < dimH(µ)
and start by few remarks.
Set
α= dim(µ)
and let
ε > 0
and
ρ > 0
be two real numbers. One denes
Eα,ρ,ε
µ=x∈Rd:∀r≤ρ, µ (B(x, r)) ≤rα−ε.
Since
µ
is
α
-exact dimensional, for
µ
-almost every
x
,
limr→0+log µ(B(x,r))
log r=
α.
This implies that, for very
ε > 0, µ ÄSρ>0Eα,ρ,ε
µä= 1.
Let
ε > 0
and
0< ρε≤1
be two real numbers such that
µ(Eα,ρε,ε
µ)≥1
2
and write
E=Eα,ρε,ε
µ.
Write
ci=|fi(K)|.
Let us x
i= (i1, ..., ik)∈Λ∗.
For any
x∈K
and

110 CHAPTER 6:
µ
-A.C SEQUENCES OF BALLS
r > 0
, by (4.35) and (4.36) applied with
θ=ε′
, one has
fi(B(x, r)) ⊃B(fi(x0),b
Cε′ci(x0)1−ε′r)⊃BÑfi(x0),b
C−2
1−ε′
ε′
|K|−1+ε′
1−ε′
c
1+ε′
1−ε′
iré.
Remember that
ε′≤1
2.
Since
1+ε′
1−ε′≤1+4ε′
,
fi(B(x, r)) ⊃BÅfi(x0),b
C−2
1−ε′
ε′· |K|1+ε′
1−ε′c1+4ε′
irã.
(6.10)
Writing
µi=µ(f−1
i)
, (6.10) yields
Ei:= fi(E)
=nfi(x)∈K:∀r≤ρε, µB(x, r)≤rα−εo
⊂ {fi(x), x ∈K:∀r≤ρε,
µf−1
iÅBÅfi(x0),b
C−2
1−ε′
ε′· |K|1+ε′
1−ε′c1+4ε′
irãã≤Ñb
C−2
1−ε′
ε′· |K|1+ε′
1−ε′c1+4ε′
ir
b
C−2
1−ε′
ε′· |K|1+ε′
1−ε′c1+4ε′
iéα−ε




=ßy∈fi(K) : ∀r′≤b
C−2
1−ε′
ε′· |K|1+ε′
1−ε′c1+4ε′
iρε,
(6.11)
µiB(y, r′)≤Ñr′
b
C−2
1−ε′
ε′· |K|1+ε′
1−ε′c1+4ε′
iéα−ε

.
Notice also that
µi(Ei) = µ(E)≥1
2.
Let us emphasize that iterating equation (4.22) gives
µ=X
i′∈Λk
pi′µi′,
which implies that
µi
is absolutely continuous with respect to
µ
(since all
pi
's
are strictly positive).
We are now ready to estimate the
µ
-essential content of a ball
B
centered
in
K
.
Let us write
γ(S, ε′) = b
C−2
1−ε′
ε′· |K|1+ε′
1−ε′.
(6.12)
Let
B=B(x, r)
with
x∈K
and
r≤c0:= minz∈Kmin1≤i≤mqf′
i(z)q
.

6.2 ESSENTIAL HAUSDORFF CONTENT 111
Since
x∈K
, there exists
i= (i1, ..., ik)∈Λ∗
such that

x∈fi(K)
,

|fi(K)| ≤ 1
3|B|
,

|f(i1,...,ik−1)(K)| ≥ 1
3|B|.
By (4.36), for any
y∈K
one has
|fi(K)| ≥ b
C−1
ε′||f′
i(y)||1+ε′|K|
(6.13)
and
||fi(y)|| =||f′
(i1,...,in−1)(fn(x)) ◦f′
in(x)|| ≥ ||f′
(i1,...,in−1)(fn(x))||c0
≥ |f(i1,...,in−1(K)|1
1−ε′b
C
−1
1−ε′
ε′· |K|−1
1−ε′c0.
(6.14)
Combining (6.13) and (6.14), one obtains
ci=|fi(K)| ≥ b
C−1−1+ε′
1−ε′
ε′|K|−2ε′
1−ε′c1+ε′
0|f(i1,...,in−1)(K)|1+ε′
1−ε′
≥b
C−1−1+ε′
1−ε′
ε′|K|−2ε′
1−ε′c1+ε′
0r1+4ε′.
(6.15)
Note that
Ei⊂˚
B
.
Consider a set
A⊂B
verifying
µ(A) = µ(B).
One aims at giving a lower-
bound for the Hausdor content of
A
which depends only on
B
,
d
,
ε
,
ε′
and
the measure
µ
.
Consider a sequence of balls
(Ln=B(xn, ℓn))n≥1
covering
A∩Ei
, such
that
ℓn< γ(S, ε′)ρεc1+4ε′
i
and
xn∈A∩Ei
.
Since
µi
is absolutely continuous with respect to
µ
, it holds that
µi(A)=1.
By (6.11) applied to each ball
Ln
,
n∈N
, one has
Å|Ln|
γ(S,ε′)c1+4ε′
iãα−ε
≥
µi(Ln)
, so that, recalling (6.15),
X
n∈N|Ln|α−ε≥X
n∈NÄγ(S, ε′)c1+4ε′
iäα−εµi(Ln)≥Äγ(S, ε′)c1+4ε′
iäα−εµi [
n∈N
Ln!
≥Äγ(S, ε′)c1+4ε′
iäα−εµi(Ei)≥1
2Äγ(S, ε′)c1+4ε′
iäα−ε
≥κ(µ, ε′, ε)r(1+4ε′)2(α−ε)≥κ(µ, ε′, ε)r(1+16ε′)(α−ε),
(6.16)
where
κ(µ, ε′, ε) = 1
2γ(S, ε′)α−ε·Åb
C−1−1+ε′
1−ε′
ε′|K|−2ε′
1−ε′c1+ε′
0ã(1+4ε′)(α−ε)
.

112 CHAPTER 6:
µ
-A.C SEQUENCES OF BALLS
This series of inequalities holds for any sequence of balls
(Ln)n∈N
with
radius less than
γ(S, ε′)ρεc1+4ε′
i
centered in
A∩Ei
. One now proves that one
can freely remove those constraints on the center and the radius of the balls
used to cover
A∩Ei
, up to a multiplicative constant.
Consider balls
(Ln=B(xn, ℓn))n≥1
covering
A∩Ei
such that
ℓn<
γ(S, ε′)ρεc1+4ε′
i
but
xn
does not necessarily belongs to
A∩Ei
.
Let
n∈N
. One constructs recursively a sequence of balls
(Ln,j)1≤j≤Jn
such that the following properties hold for any
1≤j≤Jn
:

Ln,j
is centered on
A∩Ei∩Ln
;

A∩Ei∩Ln⊂S1≤j≤JnLn,j
;

for all
1≤j≤Jn,|Ln,j|=|Ln|
;

the center of
Ln,j
does not belong to any
Ln,j′
for
1≤j′=j≤Jn
.
To achieve this, simply consider
y1∈A∩Ei∩Ln
and set
L1,n =B(y1, ℓn).
If
A∩Ei∩Ln⊈L1,n
, consider
y2∈A∩Ei∩Ln\L1,n
and set
L2,n =B(y2, ℓn)
.
If
A∩Ei∩Ln⊈L1,n ∪L2,n
, consider
y3∈A∩Ei∩Ln\L1,n ∪L2,n
and set
L3,n =B(y3, ℓn)
, and so on...
Note that, for any
1≤j≤Jn
, any ball
Lj,n
has radius
ℓn
, intersects
Ln
(which also has radius
ℓn
) and, because
yj/∈S1≤j′=j≤JnLj′,n
, it holds that, for
any
j=j′
,
1
3Ln,j ∩1
3Ln,j′=∅
. By Lemma 4.1.4, this implies that
Jn≤Qd, 1
3.
Hence, denoting by
(e
Ln)n∈N
the collection of the corresponding balls cen-
tered on
A∩Ei
associated with all the balls
Ln
, one has by (6.16) applied to
(e
Ln)n∈N
:
X
n∈N|Ln|α−ε≥1
Qd, 1
3X
n∈N|e
Ln|α−ε≥κ(µ, ε′, ε)
Qd, 1
3
r(1+4ε′)(α−ε).
Remark also that any ball of radius smaller that
ci
can be covered by at most
Å2c−4ε′
i
γ(S,ε′)ρεãd
balls of radius
γ(S, ε′)ρεc1+4ε′
i
. Moreover, by (6.15),
c−4ε′
i≤Åb
C−1−1+ε′
1−ε′
ε′|K|−2ε′
1−ε′c1+ε′
0ã−4ε′
r−4ε′·(1+4ε′).
Setting
bκ(µ, ε, ε′, d) = á2Åb
C−1−1+ε′
1−ε′
ε′|K|−2ε′
1−ε′c1+ε′
0ã−4ε′
γ(S, ε′)ρεëd
,

6.2 ESSENTIAL HAUSDORFF CONTENT 113
any ball of radius less than
ci
can be covered by less than
bκ(µ, ε, ε′, d)r−4dε′·(1+4ε′)
balls of radius less than
γ(S, ε′)ρεc1+4ε′
i
.
This proves that, for any sequence of balls
b
Ln
with
|b
Ln| ≤ ci
covering
A∩Ei
, recalling (6.16), it holds that
X
n∈N|b
Ln|α−ε≥Q−1
d, 1
3bκ(µ, ε, ε′, d)−1r4dε′·(1+4ε′)κ(µ, ε′, ε)r(1+16ε′)(α−ε)
(6.17)
≥Q−1
d, 1
3bκ(µ, ε, ε′, d)−1κ(µ, ε′, ε)r(1+16ε′)(α−ε)+4dε′·(1+4ε′).
(6.18)
Recalling that
|Ei| ≤ ci
and Denition 4.7 , since (6.17) is valid for any covering
(b
Ln)n∈N
of
A∩Ei
with
|Ln| ≤ ci
, one has
|B|α−ε≥ Hα−ε
∞(A)≥ Hα−ε
∞(A∩Ei)
≥Q−1
d, 1
3bκ(µ, ε, ε′, d)−1κ(µ, ε′, ε)r(1+16ε′)(α−ε)+4dε′·(1+4ε′).
(6.19)
Taking the inf over all the set
A⊂B
satisfying
µ(A) = µ(B)
, one obtains
|B|α−ε≥ Hµ,s
∞(B)≥Q−1
d, 1
3bκ(µ, ε, ε′, d)−1κ(µ, ε′, ε)r(1+16ε′)(α−ε)+4dε′·(1+4ε′).
The results stands for balls of diameter less than
c0.
Set
ε′
0= 16ε′(α−ε)+4dε′·(1 + 4ε′)
and write
γ(d, µ, ε, ε′
0) = cα−ε+ε′
0
0Q−1
d, 1
3bκ(µ, ε, ε′
0, d)−1κ(µ, ε′
0, ε).
For any ball of radius less than
1
centered on
K
, one has
|B|α−ε≥ Hµ,α−ε
∞(B)≥γ(d, µ, ε, ε′
0)rα−ε+ε′
0.
The estimates of Theorem 6.2.4 are now established in the case of general
open sets.
Recall that by item
(5)
of Proposition 6.2.2, for any
s > dim(µ)
and any
set
E
,
Hµ,s
∞(E) = 0.
Let us x
s < dim(µ)
,
ε′>0
and set
ε′= min ¶dim(µ)−s
2,1
2©>0.

114 CHAPTER 6:
µ
-A.C SEQUENCES OF BALLS
Since
K∩Ω⊂Ω
and
µ(K∩Ω) = µ(Ω)
, it holds that
Hµ,s
∞(Ω) ≤ Hs
∞(Ω ∩K).
It remains to show that there exists a constant
c(d, µ, s, ε′)
such that for
any open set
Ω
, the converse inequality
c(d, µ, s, ε′)Hs+ε′
∞(Ω ∩K)≤ Hµ,s
∞(Ω)
holds.
Let
E⊂Ω
be a Borel set such that
µ(E) = µ(Ω)
and
Hs
∞(E)≤2Hµ,s
∞(Ω).
(6.20)
Let
{Ln}n∈N
be a covering of
E
by balls verifying
Hs
∞(L)≤X
n≥0|Ln|s≤2Hs
∞(E).
(6.21)
The covering
(Ln)n∈N
will be modied into a covering
(e
Ln)n∈N
verifying the
following properties:

K∩Ω⊂Sn∈Ne
Ln
,

Sn∈NLn⊂Sn∈Ne
Ln

X
n≥0|e
Ln|s+ε′≤8.2s+ε′Q2
d,1
γ(d, µ, ε, ε′)X
n≥0|Ln|s,
where
Qd,1
and
γ(d, µ, ε, ε′)
are the constants arising from Proposition
4.1.2 applied with
v= 1
and Proposition 6.2.6.
Last item together with (6.20) and (6.21) then immediately imply that
γ(d, µ, ε, ε′)
8.2s+ε′Q2
d,1Hs+ε′
∞(K∩Ω) ≤ Hµ,s
∞(Ω).
Setting
c(d, µ, ε, ε′) = γ(d,µ,ε,ε′)
8.2s+ε′Q2
d,1
will then conclude the proof.
Let us start the construction of the sequence
(e
Ln)n∈N.
Let
∆ = (K\Sn∈NBn)∩Ω
. For every
x∈∆
, x
0< rx≤1
such that
B(x, rx)⊂Ω.
One of the following alternatives must occur:

6.2 ESSENTIAL HAUSDORFF CONTENT 115
1. for any ball
Ln
such that
Ln∩B(x, rx)=∅
,
|Ln| ≤ rx
, or
2. there exists
nx∈N
such that
Lnx∩B(x, rx)=∅
and
|Lnx| ≥ rx
.
Consider the set
S1
of points of
X
for which the rst alternative holds.
By Proposition 4.1.2 applied with
v= 1
, it is possible to extract from
the covering of
S1
,
{B(x, rx), x ∈S1}
,
Qd,1
families of pairwise disjoint balls,
F1, ..., FQd,1
such that
S1⊂[
1≤i≤Qd,1[
L∈Fi
L.
Now, any ball
Ln
intersecting a ball
L∈S1≤i≤Qd,1Fi
must satisfy
|Ln| ≤ L.
In particular, since for any
1≤i≤Qd,1
, the balls of
Fi
are pairwise disjoint,
applying Lemma 4.1.4 to the ball of
Fi
intersecting
L
, we get that the ball
Ln
intersects at most
Qd,1
balls of
Fi
, hence at most
Q2
d,1
balls of
S1≤i≤Qd,1Fi.
Let
L∈S1≤i≤Qd,1Fi.
One aims at replacing all the balls
Ln
intersecting
L
by the ball
2L
.
For any
1≤i≤Qd,1
and any ball
L∈ Fi
, denote by
GL
the set of balls
Ln
intersecting
L
. Since
E⊂Sn∈NLn
and
µ(E) = µ(Ω)
, one has
E∩L⊂SB∈GLB
and
µ(E∩L) = µ(L)
. By Denition 6.2.1 and Proposition 6.2.6, this implies
that
γ(d, µ, ε, ε′)|L|s+ε′≤ Hµ,s
∞(L)≤X
B∈GLHµ,s
∞(B)≤X
B∈GL|B|s.
(6.22)
Replace the balls of
GL
by the ball
b
L= 2L
(recall that
SB∈GLB⊂2L
). The
new sequence of balls so obtained by the previous construction applied to all
the balls
L∈S≤i≤Qd,1Fi
is denoted by
(b
Lk)1≤k≤K
, where
0≤K≤+∞
.
It follows from the construction and (6.22) that
S1⊂S1≤k≤Kb
Lk
and
X
1≤k≤K|b
Lk|
2s+ε′
≤Q2
d,1
γ(d, µ, ε, ε′)X
n≥0|Ln|s.
(6.23)
On the other hand, since for any
x∈S2= ∆ \S1
, there exists
nx∈N
such that
Lnx∩B(x, rx)=∅
and
rx≤ |Lnx|
, one has
S2⊂Sn∈N2Ln
, so that
[
n∈N
Ln∪K∩Ω\[
n∈N
Ln⊂[
1≤k≤Kb
Lk∪[
n∈N
2Ln.
Putting the elements of
(b
Lk)1≤k≤K
and
(2Ln)n≥0
in a single sequence
(b
Ln)n≥0
,
writing
(e
Ln:= 2b
Ln)n∈N
, by construction,
K∩Ω⊂Sn∈Ne
Ln
and due to (6.23):

116 CHAPTER 6:
µ
-A.C SEQUENCES OF BALLS
Hs+ε′
∞(K∩Ω) ≤X
n≥0|e
Ln|s+ε′≤2s+ε′Q2
d,1
γ(d, µ, ε, ε′)+ 1X
n≥0|Ln|s
≤8.2s+ε′Q2
d,1
γ(d, µ, ε, ε′)Hµ,s
∞(Ω).
The proof is concluded now by setting
c(d, µ, s, ε′) = γ(d, µ, dim(µ)−s, ε′)
Q2
d,18.2s+ε′.
Remark 6.2.7.
1. The proof of Proposition 6.2.6 only relies on the absolute
continuity, for any
i∈Λ∗
, of
µ(f−1
i)
with respect to the weakly conformal
measure
µ.
2. The part of the proof of Theorem 6.8 which handles the case of open
sets only relies on the fact that there exists
γ(d, µ, ε, ε′)
such that for
any
x∈K
, for any
ρ > 0
, there exists
0< rx≤ρ
so that, writing
B=B(x, rx),
γ(d, µ, ε, ε′)|B|dim(µ)−ε+ε′≤ Hµ,dim(µ)−ε
∞(˚
B)≤ Hµ,dim(µ)−ε
∞(B)≤ |B|dim(µ)−ε.
(6.24)
In particular Theorem 6.2.4 actually holds for any measure
µ∈ M(Rd)
for which
supp(µ)⊂K
and for any
i∈Λ∗, µ(f−1
i)
is absolutely contin-
uous with respect to
µ
(so that it holds for quasi-Bernoulli measures for
instance).

It is easily veried that the estimates of Proposition 6.2.6 holds in par-
ticular if, for
s≥0,
there exists a constant
C > 0
such that for any
x∈supp(µ)
, any
0< r < R
,
µ(B(x,r))
µ(B(x,R)) ≤C.r
Rs.
This condition is
naturally linked to the lower Assouad dimension
dimL(µ)
of
µ
dened as
[40]
dimL(µ) = inf ßs≥0 : ∀x∈supp(µ),∀0< r < R, µ(B(x, r))
µ(B(x, R)) ≤C(r
R)s™.
(6.25)
More precisely, the estimates of Proposition 6.2.6 and Theorem 6.2.4
holds for any
s < dimL(µ)
.

Chapter 7
Heterogeneous ubiquity theorem
In this Chapter, the main ubiquity theorem of this manuscript is proved. Sec-
tion 7.1 states Theorem 7.1.2. Section 7.2 is dedicated to the proof of Theorem
7.1.2. Finally in Section 7.3, Theorem 7.1.2 is applied to obtain mass transfer-
ence principles for weakly conformal measures (with no separation conditions).
7.1 A general heterogeneous ubiquity theorem
The
s
-dimensional
µ
-essential Hausdor content is now used to associate a
critical exponent to any sequence of open sets
(Un)n∈N
such that
Un⊂Bn
for all
n∈N
. This exponent is involved in our lower bound estimate of
dimH(lim supn→+∞Un)
.
Denition 7.1.1.
Let
µ∈ M(Rd)
. If
B
and
U
are Borel subsets of
Rd
, the
µ
-critical exponent of
(B, U )
is dened as
sµ(B, U ) = sup {s≥0 : Hµ,s
∞(U)≥µ(B)}.
(7.1)
Let
B= (Bn)n∈N
be a sequence of closed balls,
U= (Un)n∈N
a sequence of
Borel subsets of
Rd
, and
s≥0
.
Let
Nµ(B,U, s) = {n∈N:sµ(Bn, Un)≥s}.
(7.2)
Then, dene the
µ
-critical exponent of
(B,U)
as
s(µ, B,U) = sup s≥0 : (Bn)n∈Nµ(B,U,s)
is
µ
-a.c.
.
(7.3)
It is worth noting that, for
s′≤s
, one has
Nµ(B,U, s)⊂ Nµ(B,U, s′)
.

118 CHAPTER 7: HETEROGENEOUS UBIQUITY THEOREM
The main result of this paper is the following.
Theorem 7.1.2.
Let
B= (Bn)n∈N
be a sequence of closed balls of
Rd
such that
|Bn| → 0
and
U= (Un)n∈N
a sequence of open sets such that
Un⊂Bn
for all
n∈N
.
Then, for every
µ∈ M(Rd)
such that
min {s(µ, B,U),dimH(µ)}>0
there
exists a gauge function
ζ:R+→R+
such that
limr→0+log ζ(r)
log r= min {s(µ, B,U),dimH(µ)}
and
Hζ(lim sup
n→+∞
Un)>0.
In particular, for every
µ∈ M(Rd)
, one has
dimHÅlim sup
n→+∞
Unã≥min {s(µ, B,U),dimH(µ)}.
(7.4)
Remark 7.1.3.
(1) It is easily veried that the lower-bound in Theorem 7.1.2
equals
−∞
if the sequence
(Bn)n∈N
is not assumed to be
µ
-a.c. Consequently,
for the previous result to give non trivial information one has to assume that
(Bn)n∈N
is
µ
-a.c. The question is then to give more explicit estimates of
s(µ, B,U)
depending on the specities of
(µ, B,U)
.
(2) It is proved in Section 7.2.2 that
s(µ, B,U)≤dimH(µ).
This implies
that for exact dimensional measures,
min {s(µ, B,U),dimH(µ)}=s(µ, B,U).
(3) The case where
µ
satises
min {s(µ, B,U),dimH(µ)}= 0
could also
be treated, but although
(7.4)
is still obviously true, some distinction should
further be made when investigating the existence of the gauge function. If
Hµ,s
∞(Un)=0
for any
n∈N
, the set
lim supn→+∞Un
could, for instance,
be empty. On the other hand, if
(Bn)n∈N
is
µ
-a.c and
sµ(Bn, Un)>0
for
any
n∈N
, a gauge function can be constructed in a similar way than in the
proof of Theorem 7.1.2. However the existence of a gauge function in the case
min {s(µ, B,U),dimH(µ)}= 0
is of little interest for practical applications and
is not treated in this manuscript.
A quite direct, but useful, corollary of Theorem 7.1.2 is the following:
Corollary 7.1.4.
Let
µ∈ M(Rd)
and
B= (Bn)n∈N
be a
µ
-a.c. sequence
of closed balls of
Rd
. Let
U= (Un)n∈N
be a sequence of open sets such that
Un⊂Bn
for all
n∈N
, and
0≤s≤dimH(µ)
. If
lim supn→+∞
log Hµ,s
∞(Un)
log µ(Bn)≤1
,
then
s(µ, B,U)≥s
, so that
dimH(lim sup
n→+∞
Un)≥s.
In the classical case where the sets
Un
are shrunk balls of the form
Bδ
n

7.2 PROOF OF THEOREM 7.1.2 AND COROLLARY 7.1.6 119
(with
δ≥1
), it is convenient to consider the following quantity:
Denition 7.1.5.
Let
µ∈ M(Rd)
,
ε > 0
and
B= (Bn)n∈N
be a sequence of
balls of
Rd.
For every
δ≥1
, set
t(µ, δ, ε, B) = lim sup
n→+∞
log(Hµ,dimH(µ)−ε
∞(˚
Bδ
n))
log(|Bδ
n|).
(7.5)
Then the
(µ, δ)
-exponent of the sequence
B
is dened as
t(µ, δ, B) = lim
ε→0t(µ, δ, ε, B).
(7.6)
It follows from the denitions that
t(µ, δ, B)
exists as a limit, since
ε7→
t(µ, δ, ε, B)
is monotonic. Moreover, one has
dimH(µ)≤t(µ, δ, B)
(see the proof
of Corollary 7.1.6).
Next result provides a more explicit lower bound estimate of the Haus-
dor dimension of the limsup of
δ
-contracted balls; it is a consequence of Corol-
lary 7.1.4.
Corollary 7.1.6.
Let
µ∈ M(Rd)
and
B= (Bn)n∈N
a
µ
-a.c sequence of closed
balls of
Rd
. Suppose that
dimH(µ)>0
. For every
δ≥1
, setting
sδ=dimH(µ)
δ·dimH(µ)
t(µ, δ, B),
one has
sµ, (Bn)n∈N,(˚
Bδ
n)n∈N≥sδ,
hence
dimH(lim sup
n→+∞
Bδ
n)≥dimH(lim sup
n→+∞
˚
Bδ
n)≥sδ.
7.2 Proof of Theorem 7.1.2 and Corollary 7.1.6
7.2.0.1 Preliminary facts
We gather in this subsection a series of results on which we will base the proof
of Theorem 7.1.2.
The following lemma, which is a version of Besicovitch covering Lemma,
as well as the subsequent one, both established in [24], will be used several
times.
The following lemma will also be useful later.
Lemma 7.2.1
([24])
.
Let
L
be a family of pairwise disjoint balls satisfying
supL∈L |L|<+∞.
Then, for any
v≥1
, there exists sub-families
L1, ..., LQd,v

120 CHAPTER 7: HETEROGENEOUS UBIQUITY THEOREM
(where
Qd,v
is the constant of the same name in Proposition 4.1.2) of
L
such
that
L=S1≤i≤Qd,v Li
and for any
L=L′∈ Li
,
vL ∩vL′=∅.
For
s≥0
and
E⊂Rd
, a bounded subset such that
Hs
∞(E)>0
,
ms
E
will
always denote a measure given by Proposition 4.2.6, associated with a (xed)
constant
κd
.
In the next two lemmas, the choice of the interval
[5,6]
is convenient
to take enough space between the shrunk balls involved in the construction
elaborated in Section 7.2.1.
Lemma 7.2.2.
Let
t∈(5,6)
,
m∈ M(Rd)
, and
ε > 0
. Let
x∈Rd
be such
that
dim(m, x)≤β
. Let
Cβ,ε =1
26−β
2ε
. There exists an integer
nx
such that for
every
n≥nx
,
#0≤k≤n−1 : m(B(x, t−k−1)) ≥Cβ,ε m(B(x, t−k))
n≥1−ε.
(7.7)
Previous lemma is a slight extension of result by Käenmäki [21, Lemma
2.2], which shows such a property at
m
-almost every point (where one has
necessarily
dim(m, x)≤d)
, and uses
t
integer (a choice that we could make).
Thus, points with a given local dimension with respect to a measure
m
are for most scales locally doubling.
Proof.
Observe rst that if for a constant
0< C ≤1
and some integer
n∈N
one has
#1≤k≤n:m(B(x, t−k−1)) ≥Cm(B(x, t−k))
n≤1−ε,
then there necessarily exists
N=⌊(n−1)ε⌋
integers
0< k1<··· < kN< n
such that for every
1≤i≤N
,
m(B(x, t−ki−1)) ≤Cm(B(x, t−ki))
.
In particular, writing
kN+1 =n
and
k0= 0
, this implies that
m(B(x, t−n)) =
N
Y
i=0
m(B(x, t−ki+1 ))
m(B(x, t−ki)) ≤
N
Y
i=0
m(B(x, t−ki−1))
m(B(x, t−ki)) ≤CN
≤C(n−1)ε≤Cnε/2= (t−n)ε−log(C)
2 log(t).
The inequality
C(n−1)ε≤Cnε/2
occurs when
n
is large enough. Recalling that
dim(m, x)≤β
, if this happens for innitely many
n
, one should have
β≥lim sup
r→0+
log m(B(x, r))
log r≥ε−log(C)
2 log(t),
which is equivalent to
C≥t−β
2ε
.

7.2 PROOF OF THEOREM 7.1.2 AND COROLLARY 7.1.6 121
Setting
Cε,β =1
26−β
2ε
, one concludes that there exists
nx
such that for
every
n≥nx
, one necessarily has
#0≤k≤n−1 : m(B(x, t−k−1)) ≥Cε,βm(B(x, t−k))
n≥1−ε,
hence the result.
Lemma 7.2.3.
Let
m
and
µ
be two elements of
M(Rd)
,
β≥0
and
ε > 0
. For
every
x∈Rd
verifying
dim(m, x)≤β
, there exists
ρx>0
and
tx∈(5,6)
so
that for all
0< r ≤ρx
there exists
r≤r′≤r1−ε
such that
m(B(x, r′/tx)) ≥Cβ, ε
2m(B(x, r′))
and
µ(∂B(x, r′/tx)) = 0.
(7.8)
Proof.
Consider
x∈Rd
such that
dim(m, x)≤β.
We apply Lemma 7.2.2 to
x
and the measure
m
, and for an arbitrary
t∈[5,6]
and
ε′=ε
2
: for
n≥nx
, there must be an integer
n′
such that
n(1 −ε)≤n′≤n
and
m(B(x, t−n′−1)) ≥Cβ, ε
2m(B(x, t−n′)).
Let
ρx= min ¶t−nx−1, t−1
ε©
. For
r∈(0, ρx]
, let
n
be the integer such that
t−n−1< r ≤t−n
. The previous claim yields an integer
n′∈[n(1 −ε
2), n]
such
that
m(B(x, t−n′)) ≥Cβ, ε
2m(B(x, t−n′+1))
. Also,
r≤r′=t−n′+1 ≤t1−(1−ε
2)n=t2·t−n−1·tε
2n≤t2·r·r−ε
2≤r1−ε.
Consequently,
m(B(x, r′/t)) ≥Cβ, ε
2m(B(x, r′)).
The desired conclusion holds if we choose
tx∈(5, t)
such that
µ(∂B(x, r′/tx)) =
0
.
The previous lemma will be used in the case
β=d
in our proof the main
theorem (see step 2 of the construction in Section 7.2.1).
Next, we introduce some some sets associated to a given element of
M(Rd)
, which will play a natural role in our construction.
Denition 7.2.4.
Let
β≥α≥0
be real numbers,
m∈ M(Rd)
, and
ε, ρ > 0
two positive real numbers. Then dene
‹
E[α,β],ρ,ε
m=¶x∈Rd: dim(m, x)∈[α, β]and ∀r≤ρ, m(B(x, r)) ≤rdim(m,x)−ε©
(7.9)
and
E[α,β],ρ,ε
m=ßx∈‹
E[α,β],ρ,ε
m:∀r≤ρ, 3
4m(B(x, r)) ≤m(B(x, r)∩‹
E[α,β],ρ,ε
m)™.
(7.10)

122 CHAPTER 7: HETEROGENEOUS UBIQUITY THEOREM
Notice that, for every
0< ρ < ρ′
, one has
E[α,β],ρ′,ε
m⊂E[α,β],ρ,ε
m.
Denition 7.2.5.
Let
β≥α≥0
be real numbers,
m∈ M(Rd)
, and
ε > 0
.
Dene
E[α,β],ε
m=[
n≥1
E[α,β],1
n,ε
m.
(7.11)
Proposition 7.2.6.
For every
m∈ M(Rd)
, every
β≥α≥0
and
ε > 0
,
m(E[α,β],ε
m) = m({x: dim(m, x)∈[α, β]}).
(7.12)
Notice that, for every
0< ρ′< ρ
, one has
E[α,β],ρ,ε
m⊂E[α,β],ρ′,ε
m.
These sets play a key role in the proofs of Theorem 7.1.2 .
Proof.
Note that it is clear from Denition 4.2.12 that
{x: dim(m, x)∈[α, β]}=[
ρ>0‹
E[α,β],ρ,ε
m.
Let
ε′>0
. By Denition 4.2.12, there exists
ρε′
small enough so that
m(‹
E[α,β],ρε′,ε
m)≥(1 −ε′)m({x: dim(m, x)∈[α, β]}).
(7.13)
By Lemma 4.1.5 (and the notations therein) applied to
‹
E[α,β],ρε′,ε
m
, there exists
˜ρε′
such that
m(‹
E[α,β],ρε′,ε
m(˜ρε′)) ≥(1 −ε′)m(‹
E[α,β],ρε′,ε
m).
(7.14)
Finally for
ρ= min {ρε′,˜ρε′}
, by Denition 7.2.4 and (4.3), one has
(‹
E[α,β],ρε′,ε
m)˜ρε′⊂
E[α,β],ρ,ε
m
, so that, by (7.13) and (7.14)
m(E[α,β],ρ,ε
m)≥m((‹
E[α,β],ρε′,ε
m(˜ρε′)) ≥(1 −ε′)m(E[α,β],ε
m)
≥(1 −ε′)2m({x: dim(m, x)∈[α, β]}).
In particular
m({x: dim(m, x)∈[α, β]})≥m(E[α,β ],ε
m)≥(1−ε′)2m({x: dim(m, x)∈[α, β]}).
Letting
ε′→0
proves the result.
Corollary 7.2.7.
For every
m∈ M(Rd)
, for
α= dimH(m)
and
β= dimH(m)
,
for any
ε > 0
, one has
m(E[α,β],ε
m) = 1.
(7.15)

7.2 PROOF OF THEOREM 7.1.2 AND COROLLARY 7.1.6 123
7.2.1 Construction of the Cantor set and the measure
Recall that
µ
is a probability measure on
Rd
, and that
B= (Bn:= B(xn, rn))n∈N
is a
µ
-a.c sequence of balls of
Rd
with
limn→+∞rn= 0.
Fix
U= (Un)n∈N
a se-
quence of open sets satisfying
Un⊂Bn
for every
n∈N
.
Set
α= dimH(µ)
, and assume that
min {s(µ, B,U), α}>0
.
Our goal is to construct a gauge function
ζ:R+→R+
such that
lim
r→0+
log ζ(r)
log r= min {s(µ, B,U),dimH(µ)}
as well as
η∈ M(Rd)
supported on
lim supn→∞ Un
such that for all
r∈(0,1]
and
x∈Rd
one has
η(B(x, r)) ≤ζ(2r).
Let
(εk)k∈N
be a sequence decreasing to 0 and such that
ε1< s(µ, B,U)
.
For
k≥0
, set
sk= min {s(µ, B,U), α} − εk.
(7.16)
Along the construction of
ζ
, we only use that
sk< s(µ, B,U)
and the fact
that
sk< α
is used at the end of our analysis (see equation (7.52)).
Step 1
We need the following lemma.
Using Lemma 6.1.3 with,
(Bn)n∈Nµ(B,U,s1)
(which is
µ
-a.c since
s1<
s(µ, B,U)
),
g= 0
and
Ω = Rd
, one nds integers
N1
and
n1< ... < nN1∈
Nµ(B,U, s1)
such that :
(i) : ∀1≤i≤N1
,
Bni∩Bnj=∅
,
(ii) : µ(S1≤i≤N1Bni)≥1
2
.
By Lemma 7.2.1 applied to
{Bni}1≤i≤N1
and
v= 4
, the balls
{Bni}1≤i≤N1
can be sorted in
Qd,4
families of balls
L1, ..., LQd,4
such that

for any
1≤i≤Qd,4
, any
L=L′∈ Li
,
4L∩4L′=∅,

S1≤i≤Qd,4Li={Bni}1≤1≤N1.
At least one of these families,
Li0
, must satisfy
µ[
L∈Li0
L≥1
2Qd,4
.

124 CHAPTER 7: HETEROGENEOUS UBIQUITY THEOREM
In particular, if one must rename the balls of the family
Li0
, we can
assume that the family
{Bni}1≤i≤N1
satises
(i′) :
for any
1≤i<j≤N1
,
4Bni∩4Bnj=∅
(ii′) :
and
µ[
1≤i≤N1
Bni≥1
2Qd,4
.
(7.17)
Set
W1={Uni}1≤i≤N1
and
W1=[
1≤i≤N1
Uni.
Along the construction of the Cantor set, for every
U∈ U
, the ball of
B
naturally associated with
U
will be denoted
B[U]
(that is
B[Un]=Bn
).
The pre-measure
η
on the
σ
-algebra generated by the sets of
W1
is dened
by
for every
U∈ W1
,
η(U) = µ(B[U])
P‹
U∈W1µ(B[‹
U]).
(7.18)
It is obvious that
η(Rd) = η(W1) = 1
.
Recalling (7.2) and (7.3), since
s1< s(µ, B,U)
, the sub-sequence
(Bn)n∈Nµ(B,U,s1)
is
µ
-a.-c. Remember also that
limn→+∞rn= 0
and for every
n∈N
,
|Un| ≤ rn.
So, for every
n∈ Nµ(B,U, s1)
,
Hµ,s1
∞(Un)≥µ(Bn)
and
|Un| ≤ rn.
(7.19)
In particular, by Denition 6.2.1, for every
n∈ Nµ(B,U, s1)
for any set
En⊂Un
with
µ(En) = µ(Un)
,
µ(Bn)≤ Hµ,s1
∞(Un)≤ Hs1
∞(En).
By Proposition 4.2.6, and the notations therein, one has
ms1
En(Un)=1≤κd|Un|s1
Hs1
∞(En)≤κd|Un|s1
µ(Bn).
This implies that
µ(Bn)≤κd|Un|s1.
(7.20)
By equation (7.20), recalling the fact that the sets
W1⊂ {Un}n∈N
, one
has for every
U∈ W1,
η(U)≤µ(B[U])
1
2Qd,4≤2Qd,4κd|U|s1.
(7.21)

7.2 PROOF OF THEOREM 7.1.2 AND COROLLARY 7.1.6 125
Step 2
This step (and all the following steps) is split into two sub-steps. First, into
each open set
U
of
W1
, smaller intermediary balls are selected according to the
µ
-essential content of
U
. Then in a second time, each intermediary ball will be
covered by balls of the sequence
(Bn)n∈N
according to the measure
µ
and, as in
step 1, the sets
Un
associated with this covering will form the generation
W2
.
Let
g∈N
be such that for every
n≥g
,
rn≤1
3min(|U|:U∈ W1)
.
As above, since
s2< s(µ, B,U)
, the sub-sequence
(Bn)n∈Nµ(B,U,s2),n≥g
is
µ
-a.c. The same arguments as above yield for every
n∈ Nµ(B,U, s2)
,
Hµ,s2
∞(Un)≥µ(Bn)
and
|Un| ≤ rn
(7.22)
and
µ(Bn)≤κd|Un|s2.
(7.23)
Covering with respect to the
µ
-essential content
Consider
U∈ W1
. Set
β= dimH(µ)
. For
0≤k≤ ⌊β−α
ε2⌋+ 1
, dene
θk=
α+kε2
. Write
EU=U∩E[α,β],ε2
µ∩lim sup
n→+∞
Bn.
(7.24)
Notice that by Proposition 7.2.6 and by item (1) of Theorem 6.1.2, one has
µ(EU) = µ(U)
.
In addition, using the denition (6.7) of
Hµ,s2
∞
, the fact that
EU⊂U
and
µ(EU) = µ(U)
, and nally (7.1) applied with
Bn=B[U]
, one gets
Hs2
∞(EU)≥ Hµ,s2
∞(U)≥µ(B[U])>0.
(7.25)
This allows us to apply Proposition 4.2.6: there exists a Borel probability
measure
ms2
EU
supported on
EU
such that for every ball
B:= B(x, r)
, one has
ms2
EU(B)≤κd
rs2
Hs2
∞(EU).
Also, since
ms2
EU(EU)=1
and
EU⊂E[α,β],ε2
µ
, and recalling (7.11), for any
0≤k≤ ⌊β−α
ε2⌋+ 1
, there exists
ρk,ε2
such that
ms2
EU(E[θk,θk+1],ρk,ε2,ε2
µ)≥1
2ms2
EU(E[θk,θk+1],ε
µ).

126 CHAPTER 7: HETEROGENEOUS UBIQUITY THEOREM
Setting
ρU= min0≤k≤⌊β−α
ε2⌋+1 ρk,ε2
one has, for any
0≤k≤ ⌊β−α
ε2⌋+ 1
,
ms2
EU(E[θk,θk+1],ρU,ε2
µ)≥1
2ms2
EU(E[θk,θk+1],ε
µ).
(7.26)
In particular,
ms2
EU(E[α,β],ρU,ε2
µ)≥1
2
(7.27)
Let
SU:= [
0≤k≤⌊β−α
ε2⌋+1
E[θk,θk+1],ρU,ε2
µ∩EU∩x∈Rd: dim(ms2
EU, x)≤d.
(7.28)
Recalling that for every probability measure
m
,
m({x= dim(m, x)≤d})=1
,
one necessarily has
ms2
EU(SU)≥1/2
.
Let
x∈SU
; consider
0≤kx≤ ⌊β−α
ε2⌋+ 1
such that
x∈E[θkx,θkx+1],ρU,ε2
µ.
Applying Lemma 7.2.3, there exists
0< rx<min ρx,1
3min {|V|:V∈ W1}
and
tx∈(5,6)
such that:
10 rx< ρU;
(7.29)
B(x, rx)⊂U
and
µ(∂B(x, rx/tx)) = 0;
(7.30)
r−ε2
x≥5d4Qd,1
Cε3,d
η(U)
µ(B[U])≥5s24Qd,1
Cε2,d
η(U)
µ(B[U]);
(7.31)
rθkx+2ε2
x≤µ(B(x, rx)) ≤rθkx−2ε2
x;
(7.32)
ms2
EU(B(x, rx/tx)) ≥Cε2,d ·ms2
EU(B(x, rx)).
(7.33)
Note that in (7.31) the second inequality follows automatically from the rst
one since
s2≤α≤d
and the constant
Cε,d
is an increasing function of
ε
.
The family
{B(x, rx) : x∈SU}
forms a covering of
SU
. We apply Propo-
sition 4.1.2 with
v= 1
(i.e., the standard Besicovich covering Theorem) to this
family to extract
Qd,1
subfamilies of balls,
GU
1, ..., GU
Qd,1
such that:

∀1≤i≤Qd,1
,
∀B=B′∈ GU
i
, one has
B∩B′=∅,

SU⊂SQd,1
i=1 SB∈GU
iB.
In particular,
ms2
EUÄSQd,1
i=1 SB∈GU
iBä≥ms2
EU(SU)≥1/2
.
At least one of these families, say
GU
i0
, veries that
ms2
EUÖ[
B∈GU
i0
Bè≥ms2
EU(SU)
Qd,1≥1
2Qd,1
.

7.2 PROOF OF THEOREM 7.1.2 AND COROLLARY 7.1.6 127
Writing
GU
i0=BU
i0,kk∈N
, one can nd an integer
NU
so large that
ms2
EU [
1≤k≤NU
BU
i0,k!≥1
4Qd,1
.
Remind that each
BU
i0,k
is a ball
B(x, rx)
satisfying (7.30), (7.31) and
(7.33).
Finally, setting
GU=B(x, rx/tx) : B(x, rx)∈ FU
i0
, one has by construc-
tion
ms2
EU [
B∈GU
B!=X
B∈GU
ms2
EU(B)≥Cε2,d
4Qd,1
.
(7.34)
One then extends the pre-measure
η
to the Borel
σ
-algebra generated by
the balls of
GU
, by the formula
for every
B∈ GU
,
η(B) = η(U)×ms2
EU(B)
PB′∈GUms2
EU(B′).
(7.35)
By construction, this formula is consistent since
η(U) = PB∈GUη(B)
.
Observe that by (4.10), (7.34) and (7.25), one has for every
B∈ GU
,
η(B)≤η(U)κd|B|s2
Hs2
∞(EU)
4Qd,1
Cε2,d ≤4Qd,1κd
Cε2,d
η(U)
µ(B[U])|B|s2≤ |B|s2−ε2,
(7.36)
where the second inequality of (7.31) was used.
This is achieved simultaneously for all
U∈ W1
.
Covering with respect to
µ
Now, in order to build the second generation of the Cantor set
K
, we select
balls of
B
that lie in the interior of these intermediate balls
B∈ GU
.
Let
U∈ W1
and
B∈ GU
be one of these intermediary balls. Since
B
is
µ
-a.c., the last part of Lemma 6.1.3 proves the existence of a nite family
FB={Uni}1≤i≤NB
such that
(
i1
) for every
1≤i≤NB
, one has
Bni⊂˚
B
and
max ß2Qd,4
η(B)
µ(B),5d4Qd,1κd
Cε3,d ™≤r−ε2
ni,
(7.37)
(
i2
) for every
1≤i=j≤NB
, one has
Bni∩Bnj=∅.

128 CHAPTER 7: HETEROGENEOUS UBIQUITY THEOREM
In addition, recalling that
µ(∂B)=0
by (7.30), one has
µ(Bni)>0
and
µ[
1≤i≤NB
Bni≥3µ(˚
B)
4=3µ(B)
4.
Recall the denitions (7.10) and (7.28) of the sets
E[a,b],ρU,ε2
µ
and
SU
. By
equations (7.29)-(7.33), there exists
α≤a≤β
such that the center of
B
belongs to
SU⊂E[a,a+ε2],ρUε2,
µ
and
|B| ≤ ρU
, hence one has
µ(B∩‹
E[a,a+ε2],ρU,ε2
µ)≥3
4µ(B).
By
(i2)
, and recalling (7.10), one has
µ[
Bni:Bni∩‹
E[a,a+ε2],ρU,ε2
µ=∅
Bni≥µ[
1≤i≤NB
Bni∩‹
E[a,a+ε2],ρU,ε2
µ
=µ [
1≤i≤NB
Bni!+µÄ‹
E[a,a+ε2],ρU,ε2
µä−µ ‹
E[a,a+ε2],ρU,ε2
µ[ [
1≤i≤NB
Bni!
≥3
4µ(B) + 3
4µ(B)−µ(B) = 1
2µ(B).
By a slight abuse of notations, up to an extraction, we still denote by
{Bni}1≤i≤NB
the balls
Bni
such that
Bni∩‹
E[a,a+ε2],ε2,ρU
µ=∅
. The last inequality implies that
the family of balls
{Bni}1≤i≤NB
can be chosen so that it veries conditions
(i1)
and
(i2)
, as well as the two following additional conditions:
(i3)µ(Bni)>0
and
µ[
1≤i≤NB
Bni≥µ(˚
B)
2=µ(B)
2,
(i4)
for every
1≤i≤NB, Bni∩‹
E[a,a+ε2],ρU,ε2
µ=∅.
The obtained family is still denoted by
FB
.
Applying again Lemma 7.2.1 to
FB
with
v= 4
, as in step one (see (7.17),
(i′)
and
(ii′)
), if one must consider a subfamily, one can assume that the family
FB
satises
(i1)
and
(i4)
as well as the following condition
(i′
2)
and
(i′
3)
:
(i′
2) :
for every
1≤i=j≤NB
, one has
4Bni∩4Bnj=∅.
(i′
3) : µ(Bni)>0
and
µS1≤i≤NBBni≥µ(˚
B)
2Qd,4=µ(B)
2Qd,4,
Finally one denes
W2=[
U∈W1[
B∈F1,U FB
and
W2=[
L∈W2
L.

7.2 PROOF OF THEOREM 7.1.2 AND COROLLARY 7.1.6 129
The pre-measure
η
is then extended to the
σ
-algebra generated by the
elements of
W2
by setting for every
U∈ W1
, every
B∈ GU
and
V∈ FB
,
η(V) = η(B)×µ(B[V])
PV′∈FBµ(B[V′]).
(7.38)
By construction, one has
PV∈FBη(V) = η(B)
. Also, (7.37),(7.38), and
(i′
3)
imply
η(V)
µ(B[V])≤2Qd,4
η(B)
µ(B)≤ |V|−ε2,
(7.39)
so that by (7.23) and (7.39) one has
η(V)≤2Qd,4
η(B)
µ(B)×µ(B[V])≤ |B[V]|−ε2|V|s2≤ |V|s2−ε2.
(7.40)
7.2.1.1 Recurrence scheme and end of the construction
Let
p∈N∗
be an integer, and set
W0=Rd
. Suppose that sets of balls
W1
, ...,
Wp
as well as the measure
η
are constructed such that :
1. for every
1≤q≤p
,
Wq⊂ {Un}n≥q
,
Wq⊂ Wq−1
, and
η
is dened on the
σ
-algebra generated by the elements of
Sp
q=1 Wq
.
2. For every
1≤q≤p−1
, for every
U∈ Wq
, setting, as in step 2,
EU= lim supn∈Nµ(B,U,sq)Bn∩U∩E[α,β],εq
µ
, then
Hsq
∞(EU)>0
. If
msq
EU
stands for the measure associated with
EU
provided by Proposition 4.2.6,
there exists
ρU>0
such that for every
0≤k≤ ⌊β−α
εq⌋+ 1
, setting
θk=θ(q)
k=α+kεq
, one has
msq
EU(EU∩E[θk,θk+1],ρU,εq
µ)≥1
2msq
EU(EU∩E[θk,θk+1],εq
µ).
In particular,
msq
EU(EU∩[
0≤k≤⌊β−α
εq⌋+1
E[θk,θk+1],ρU,εq
µ)≥1
2.
3. For every
1≤q≤p−1
, for every
U∈ Wq
, there exists a nite family
GU
of balls
B(x, rx/tx)
, where
x
,
rx<1
3min ¶|e
U|:e
U∈ Wq©
and
tx
satisfy
(7.29), (7.30), (7.31), (7.33) and (7.34). Also, if
B=B′∈ GU
,
3B∩3B′=
∅
.
Also, for every
B∈ GU
, (7.35) and (7.36) hold true. Moreover
Wq+1 ⊂
SU∈WqGU
.

130 CHAPTER 7: HETEROGENEOUS UBIQUITY THEOREM
4. For every
1≤q≤p−1
, for every
U∈ Wq
, for every
B∈ GU
there exists
a family
FB⊂ {Un}n≥q
of pairwise disjoint open sets such that :

for every
e
U=b
U∈ FB
, one has
4B[‹
U]∩4B[“
U]=∅;
(7.41)

for every
e
U∈ FB
,
e
U⊂˚
B
, (7.38) and (7.40) hold true, as well as
2Qd,4
η(B)
µ(B)≤ |B[‹
U]|−εq+1
(7.42)
and
B[‹
U]∩‹
E[θkB,θkB+1],ρU,εq+1
µ=∅;
(7.43)

the following inequality also holds true:
µÑ[
‹
U∈FB
B[‹
U]é≥µ(B)
2Qd,4
.
(7.44)
In item (3), the fact that
3B∩3B′=∅
just follows from the choice of
B(x, rx/tx)
instead of simply
B(x, rx)
.
The proof follows then exactly and rigorously the same lines as those of
Step 2. We do not reproduce it here, the only dierences are that
W1
,
W2
and
s2
are replaced by
Wp
,
Wp+1
and
sp+1
.
Finally, dene the Cantor set
K=\
p≥1
Wp=\
p≥1[
V∈Wp
B[V].
Applying Caratheodory's extension Theorem to the pre-measure
η
yields a
probability outer-measure on
Rd
that we still denote by
η
, which is metric,
so that Borel sets are
η
-measurable and its restriction to Borel sets belongs
to
M(Rd)
. The so obtained measure
η
is fully supported on
K
. Also, for
every
p≥2
, for any
U∈ Wp
,
B∈ GU
, and
e
U∈ FB
, the inequalities (7.35),
(7.36),(7.38) and (7.40) holds with
sp
and
εp
instead of
s2
and
ε2
.
7.2.1.2 Upper-bound for the mass of a ball
Dene the gauge function
ζ:R+7→ R+
as follows:

if for some
p≥1
,
1
3min {|U|:U∈ Wp+1} ≤ r < 1
3min {|U|:U∈ Wp}
,
then
ζ(r) = 2Qd,410drsp−5εp
,

7.2 PROOF OF THEOREM 7.1.2 AND COROLLARY 7.1.6 131

if
r≥1
3min {|U|:U∈ W1}
,
ζ(r) = 1
,

ζ(0) = 0
.
Since
εp→0
, one checks that
limr→0+log(ζ(r))
log(r)= min {s(µ, B,U),dimH(µ)}
.
Let
A
be a ball of radius
r
. If there exists
n∈N
such that
A
does not
intersect
Kn
then
η(A) = η(A∩Kn)=0.
Suppose that for every
n∈N
,
A
intersects
Kn
. The goal is to prove that
η(A)≤ζ(|A|)
when
|A|
is small.
Some cases must be distinguished.
First if for every
n∈N
,
A
intersects only one contracted set
Vn
of
Kn
,
then by (7.36)
η(A)≤η(Vn)≤ |Vn|sn−εn→
n→+∞0.
In the other case, there exists
p∈N
such that
A
intersects only one
element of
Wp
, and at least two elements of
Wp+1
. Denote by
U
the unique
element of
Wp
intersecting
A
.
1.
Case 1:
If
|A|≥|U|
, then by (7.40)
η(A)≤η(U)≤ |U|sp−εp≤ζ(|A|).
(7.45)
2.
Case 2:
If
|A|<|U|
and
A
intersects at least two balls of
GU
: Observe
that when
A
intersects two balls
B
and
B′
of
GU
, since by item
(3)
of the
recurrence scheme
3B∩3B′=∅
, one necessarily has (by Lemma 5.2.1)
B∪B′⊂5A
. Hence,
SB∈GU:B∩A=∅B⊂5A
and by (7.35) and (7.34),
η(A) = η(U)×PB∈GU:B∩A=∅msp+1
EU(B)
PB′∈GUmsp+1
EU(B′)≤4Qd,1
Cεp+1,d
η(U)msp+1
EU(5A).
Then, by (4.10), (7.25), (7.37) and (7.39)
η(A)≤4Qd,1
Cεp+1,d
η(U)κd
(5|A|)sp+1
Hµ,sp+1
∞(EU)≤5sp+1 4Qd,1κd
Cεp+1,d
η(U)
µ(B[U])|A|sp+1
≤ |A|sp+1 |U|−2εp≤ |A|sp+1−2εp+1 ≤ζ(|A|),
(7.46)
where we used that
|A|<|U|
, and the mappings
x7→ |U|−x
and
x7→
x−εp+1
are decreasing.
3.
Case 3:
If
A
intersects only one ball of
GU
: calling
B
this particular
ball and
rB
its radius (at this stage there should be no confusion with
the radii of the terms of the sequence
(Bn)n≥1
), two cases must again be
distinguished:

132 CHAPTER 7: HETEROGENEOUS UBIQUITY THEOREM
(a)
Subcase 3.1:
|B|≤|A|
:
by (7.36),
η(A)≤η(B)≤ |B|sp+1−εp+1 ≤ |A|sp+1 −εp+1 ≤ζ(|A|).
(7.47)
(b)
Subcase 3.2:
|A| ≤ |B|
:
Denote by
kB
the integer such that its
center belongs to
E[θkB,θkB+1],ρU,εp+1
µ
.
The ball
A
must intersect at least two elements
V=V′
of
Wp+1
(by
denition of
p
). Note that those sets must belong to
FB
(because
A
intersects only
B
). Applying Lemma 5.2.1 to the ball
A
with any
of those ball
V∈ Fp+1
, since
A∩V=∅
and
A\B[V]=∅
(because
A
intersects an other dilated ball,
B[V′]
by hypothesis and two such
balls veries (7.41)), one has
[
V∩A=∅
B[V]⊂5A.
(7.48)
Then, (7.38) and (7.44) imply that
η(A) = η(B)·PV∈Wp+1:V∩A=∅µ(B[V])
PV′∈FBµ(B[V′])≤2Qd,4
η(B)
µ(B)µ(5A).
(7.49)
Recalling (7.48), the ball
5A
contains some of the balls of
FB
: Hence,
by (7.43),
‹
E[θkB,θkB+1],ρU,εp+1
µ∩5A=∅
. Since
|A|≤|B|
, by (7.29),
since
rB<1
10 ρU
, for any
x∈‹
E[θkB,θkB+1],ρU,εp+1
µ∩5A
one has
µ(5A)≤µ(B(x, 10r)) ≤(10r)θkB−2εp+1 .
(7.50)
Recalling (7.32) (applied to the ball
B
), one has
µ(B)≥(rB)θkB+2εp+1 .
(7.51)
Using (7.36) (applied to
B
) (7.49), (7.50) and (7.51), one obtains
η(A)≤2Qd,4rsp+1−εp+1
B10rθkB−2εp+1
rθkB+2εp+1
B
= 2Qd,410θkB−2εp+1 rsp+1−θkB−εp+1
δ−2εp+1
B
rsp+1−θkB−εp+1 −2εp+1 rsp+1−θkB−εp+1 −4εp+1
≤2Qd,410θkB−2εp+1 rsp+1−5εp+1 .
Finally, recalling (7.16),
sp+1 −5εp+1 ≤α≤θk
, and since
rB≥r

7.2 PROOF OF THEOREM 7.1.2 AND COROLLARY 7.1.6 133
and
sp≤sp+1
, one gets
η(A)≤2Qd,410θk−εp+1 rsp+1−5εp+1 ≤2Qd,410d−εp+1 |A|sp−5εp
≤2Qd,410d−εp+1 |A|sp−5εp,
hence
η(A)≤ζ(|A|).
(7.52)
Since for any
p∈N
and any ball
A
satisfying
|A| ≤ 1
3min {|U|:U∈ Wp}
, if
A
intersects at most one element of
Wp
, the inequalities (7.45), (7.46), (7.47),
(7.52) proves that for any such ball, one has
η(A)≤ζ(|A|)
.
Hence recalling Denition 4.2.2, by the mass distribution principle, one
deduces that
Hζ(K)≥1
, which concludes the proof of Theorem 7.1.2.
7.2.2 Proof of Corollary 7.1.6
7.2.2.1 Proof of Corollary 7.1.6
One starts with a lemma.
Lemma 7.2.8.
Let
µ∈ M(Rd)
and
B= (Bn:= B(xn, rn))n∈N
be a
µ
-a.c
sequence of balls of
Rd
with
limn→+∞rn= 0
.
For any
ε > 0
, there exists a
µ
-a.c subsequence
(Bϕ(n))n∈N
of
B
such that
for every
n∈N
,
µ(Bϕ(n))≤(rϕ(n))dimH(µ)−ε.
Proof.
Set
α= dimH(µ)
and
γ=suppessµ(dim(µ, x)).
Let
Ω
be an open set and
ε > 0
. By (7.2.6),
µ(E[α,γ],ε
2
µ) = 1
and
µ(Ω ∩E[α,γ],ε
2
µ) = µ(Ω).
For every
x∈Ω∩E[α,γ],ε
2
µ
, there exists
rx>0
such that
B(x, rx)⊂Ω
and
x∈E[α,γ],rx,ε
2
µ.
Recall (7.11) and that the sets
E[α,γ],ρ,ε
µ
are non-increasing in
ρ
. In partic-
ular there exists
ρΩ>0
such that the set
EΩ:= nx∈Ω∩E[α,γ],ρΩ,ε
2
µ:rx≤ρΩo
veries
µ(EΩ)≥3µ(Ω)
4.
(7.53)
Let
g∈N
. Applying Lemma 6.1.3 to
Ω
, the sequence
(Bn)
and the
measure
m
, there exists
NΩ
as well as
g≤n1≤... ≤nNΩ
verifying:
1. for every
1≤i=j≤NΩ, Bni∩Bnj=∅
,

134 CHAPTER 7: HETEROGENEOUS UBIQUITY THEOREM
2. for every
1≤i≤NΩ
,
2rni≤ρΩ
and
2α−ε
2≤r−ε
2
ni
,
3.
µ(S1≤i≤NΩBni)≥µ(Ω)
2.
We may assume that
µ(Bni)>0
for every
i
, otherwise
Bni
does not play any
role.
Item
(3)
together with (7.53) implies that
µ [
1≤i≤NΩ
Bni∩EΩ!≥µ(Ω)
4.
Furthermore, for every
1≤i≤NΩ
verifying
Bni∩EΩ=∅
, it holds that
0< µ(Bni)≤(rni)α−ε.
Indeed, let
x∈Bni∩EΩ
. By item (2),
Bni⊂B(x, 2rni)
,
and by (7.9) , item
(2)
, and (7.2.7), it holds that
µ(Bni)≤µ(B(x, 2rni)) ≤(2rni)α−ε
2≤(rni)α−ε.
Writing
B′={Bn:µ(Bn)≤rα−ε
n}
, the argument above shows that only
balls of
B′
have been used to cover
Ω
. This is satised for every open set
Ω
,
so that
B′
is a sub-sequence of
B
satisfying the condition of Denition 6.1.1,
which concludes the proof of Lemma 7.2.8.
Proof of Corollary 7.1.6.
(1) Observe that item (2) of Proposition 6.2.2 implies
that
t(µ, δ, ε, B)≥dimH(µ)−ε
, and
t(µ, δ, B)≥dimH(µ)
.
Now choose
ε > 0
so small that
(1−ε)dimH(µ)
δ·t(µ,δ,ε,B)≤1
. Recalling Lemma 7.2.8,
up to an extraction, one can assume that for any
n∈N
,
µ(Bn)≤ |Bn|(1−ε2).dimH(µ).
Due to (7.5), there exists
Nε∈N
such that for any
n≥Nε
,
Hµ,dimH(µ)−ε
∞(˚
Bδ
n)≥ |Bδ
n|(1+ε).t(µ,δ,ε,B).
Then, Proposition 6.2.2 (4) implies that for every
n≥Nε
,
Hµ, (1−ε)dimH(µ)×(dimH(µ)−ε)
δ·t(µ,δ,ε,B)
∞(˚
Bδ
n)≥(Hµ,dimH(µ)−ε
∞(Bδ
n))
(1−ε)dimH(µ)
δ·t(µ,δ,ε,B)
≥ |Bδ
n|(1+ε)t(µ,δ,ε,B)
δ·t(µ,δ,ε,B)(1−ε).dimH(µ)
≥ |Bn|(1+ε)(1−ε)dimH(µ)≥µ(Bn).

7.3 UBIQUITY THEOREMS AND SELF-CONFORMAL MEASURES135
Thus, setting
sδ,ε =(1−ε)dimH(µ)×(dimH(µ)−ε)
δ·t(µ,δ,ε,B)
, Corollary 7.1.4 yields
dimH(lim sup
n→+∞
˚
Bδ
n)≥sδ,ε.
Since the result holds for any
ε > 0
, one gets the desired conclusion.
7.3 Ubiquity theorems and self-conformal mea-
sures
Combining Theorem 7.1.2 and Corollary 7.1.6 with Theorem 6.2.4 and Theorem
6.1.2 yield the following results for weakly conformal measures.
7.3.0.1 Mass transference principle from ball to ball
Theorem 7.3.1.
Let
S
be a weakly conformal IFS of
Rd
with attractor
K
and
µ
be a weakly conformal measure associated with
S
. Let
(Bn)n∈N
be a sequence
of closed balls centered on
K
, such that
limn→+∞|Bn|= 0
.
1. Suppose that
(Bn)n∈N
is
µ
-a.c. Then
tµ, δ, (Bn)n∈N≤dim(µ)
; conse-
quently
sµ, (Bn)n∈N,(˚
Bδ
n)n∈N≥dim(µ)
δ
and there exists a gauge func-
tion
ζ
such that
limr→0+log(ζ(r))
log(r)≥dim(µ)
δ
and
Hζ(lim supn→∞ ˚
Bδ
n)>0
. In
particular
dimH(lim sup
n→+∞
˚
Bδ
n)≥dim(µ)
δ.
(7.54)
2. Suppose that
µ(lim supn→+∞Bn)=1
. Then,
(7.54)
still holds but the ex-
istence of the gauge function is not ensured. Furthermore if
µ
is doubling,
then
(Bn)n∈N
is
µ
-a.c, so that the conclusion of item (1) holds.
Let
µ
be a weakly similar measure with support
K
, and set
α= dim(µ).
Let
(Bn:= B(xn, rn))n∈N
be a sequence of balls such that
xn∈K
for all
n∈N
,
limn→+∞rn= 0
and
µ(lim supn→+∞Bn) = 1.
Fix
ε > 0
,
v > 1
and
δ≥1
and set
Bv={vBn}n∈N
. Theorem 6.1.2 shows
that
Bv
is
µ
-a.c. Then, by Proposition 6.2.6, for
n
large enough, one has
Hµ,α−ε
∞(˚
(vBn)δ)≥κ(d, µ, ε)(vrn)δ(α−ε)≥(vrn)δ(α−ε
2).
Consequently,
t(µ, δ, ε, Bv) = lim sup
n→+∞
log Hµ,α−ε
∞(˚
(vBn)δ)
δlog |vBn|≤α−ε
2

136 CHAPTER 7: HETEROGENEOUS UBIQUITY THEOREM
so
t(µ, δ, ε, Bv)≤α.
Due to Corollary 7.1.6, one concludes that
dimH(lim sup
n→+∞
(vBn)δ)≥α
δ.
But for any
ε′>0
,
lim supn→+∞(vBn)δ⊂lim supn→+∞Bδ−ε′
n
, so that
dimH(lim sup
n→+∞
Bδ−ε′
n)≥α
δ.
It follows that for any
ε′>0
and
δ≥1
one has
dimH(lim sup
n→+∞
Bδ
n)≥α
δ+ε′.
Letting
ε′→0
proves that
dimH(lim supn→+∞Bδ
n)≥dim(µ)
δ
, hence the result.
Remark 7.3.2.

If the sequence of balls
(Bn)n∈N
is not assumed to be
µ
-a.c, but only to ver-
ify
µ(lim supn→+∞Bn)=1
, then the same lower-bound estimate holds for
dimH(lim supn→∞ Bδ
n)
, but the existence of a gauge function as in Theorem
7.1.2 does not hold in general.

Let us also notice that the computation in the proof of Theorem 7.3.1 actually
shows that, under the assumption that
limn→+∞log µ(Bn)
log |Bn|= dim(µ)
, it holds that,
for
n
large enough,
Hµ,s
∞(Bδ
n)≥µ(Bn)⇔s < dim(µ)
δ.
One also emphasizes that Theorem 7.1.2 allows to deal with very shrunk
balls as soon as the measure has positive lower Assouad dimension.
Proposition 7.3.3.
Let
µ∈ M(Rd)
and
(Bn)n∈N
be a sequence of balls veri-
fying
limn→+∞|Bn|= 0
and
µ(lim supn→+∞Bn) = 1.
Then, for any
δ≥1
1. if
δ≥dimH(µ)
dimL(µ)
, then
dimH(lim sup
n→+∞
Bδ
n)≥dimH(µ)
δ.
2. if
δ≤dimH(µ)
dimL(µ)
, then
dimH(lim sup
n→+∞
Bδ
n)≥dimL(µ).
Proof.
Let
µ∈ M(Rd)
such that
β:= dimL(µ)>0
and set
α:= dimH(µ)
.
(1) :
Fix
δ > α
β
,
ε > 0
such that
α−ε
2
βδ ≥1
and
v > 1
.

7.3 UBIQUITY THEOREMS AND SELF-CONFORMAL MEASURES137
Let us set
Bv={vBn}n∈N.
Theorem 6.1.2 shows that
Bv
is
µ
-a.c.
By Lemma 7.2.8, up to an extraction, one can assume that for any
n∈N
,
µ(vBn)≤(v|Bn|)α−ε.
By applying item (4) of Proposition 6.2.2, Proposition 6.2.6 together with
Remark 6.2.7, for every
n∈N
large enough,
Hα−ε
2
δ(◦
(vBn)δ)≥(Hβ(◦
(vBn)δ))
α−ε
2
βδ
≥C
α−ε
2
βδ (vδβ |Bn|βδ)
α−ε
2
βδ
=C
α−ε
2
βδ (v|Bn|)α−ε
2≥(v|Bn|)α−ε≥µ(vBn).
Then,
ε
being arbitrary, this proves that ( as in the proof of 7.3.1) that
s(µ, δ, Bv)≥α
δ
. Since for any
ε′
,
lim supn→+∞(vBn)δ+ε′⊂lim supn→+∞Bδ
n
,
one concludes that
dimH(lim sup
n→+∞
Bδ
n)≥α
δ+ε′.
Letting
ε′→0
proves the result in case
(1).
(2) :
Fix
δ < α
β
. Let
ε > 0
such that
δβ < α −ε
2.
By Lemma 7.2.8, one can assume once again that for any
n∈N
,
µ(vBn)≤
(v|Bn|)α−ε.
Using again Proposition 6.2.6 together with remark 6.2.7, one has
Hµ,β
∞
◦
(vBn)δ≥(v|Bn|)δβ ≥(v|Bn|)α−ε
2≥µ(vBn).
In particular this proves item
(2)
for
δ < α
β.
Let
εk→0.
Applying item
(2)
with
δ=α
β−εk
proves that item
(2)
still
stands for
δ=α
β.
Remark 7.3.4.

Following the notation of Denition 6.25, one has in
particular, for any
x∈supp(µ)
,
dim(µ, x)≥dimL(µ)
. In particular, this
implies
dimH(µ)≥dimL(µ).

Let
µ∈ M(Rd)
with
dimL(µ)>0
and
(Bn)n∈N
a
µ
-a.c sequence of balls
of
Rd
. Theorem 7.3.3 implies in particular one gets the expected lower

138 CHAPTER 7: HETEROGENEOUS UBIQUITY THEOREM
bound for
lim supn→+∞Bδ
n
for any
δ
large enough (independently of the
sequence
(Bn)n∈N
).
One emphasizes that the there isn't any particular reason for which the
Assouad dimension should be a good notion of dimension for ubiquity. In
particular, in Theorem 7.3.3, the bound when the lower bound is
dimL(µ)
,
in most cases, this lower bound could be not accurate.
7.3.0.2 Mass transference principle from ball to rectangle
One emphasizes again that
S
is not assumed to verify any separation condition.
Theorem 7.1.2 and Theorem 6.2.4 actually allows to deal with more gen-
eral sets than shrunk balls. For instance, assume that, under the hypothesis
of Theorem 7.3.1, the open sets
Un⊂Bn
are shrunk rectangles inside of
Bn
of length-sides
Qd
i=1 |Bn|τi,
where
1≤τ1≤... ≤τd.
Denote
C(Un)
the dyadic
cubes
C⊂Un
of length-sides
≈ |Bn|τd
intersecting
K
. Assume now that, for
some
s≥0,
for any
n∈N
large enough,
Hs
∞SC∈C(Un)C⪆|Bn|α,
then one
gets
dimH(lim supn→+∞Un)≥s.
In particular, when the attractor
K
is the closure of its interior, one
obtains the following extension of Theorem 5.1.5.
Denition 7.3.5.
Let
1≤τ1≤... ≤τd
and
τ= (τ1, ..., τd)
. For any
x=
(xi)1≤i≤d∈Rd
and
r > 0
, the
τ
-rectangle centered in
x
and associated with
r
is dened by
Rτ(x, r) =
d
Y
i=1
[xi−1
2rτi, xi+1
2rτi].
(7.55)
Theorem 7.3.6.
Let
S
be a weakly conformal IFS of
Rd
such that the attractor
K
is equal to the closure of its interior. Let
µ
be a weakly conformal measure as-
sociated with
S
. Let
1≤τ1≤... ≤τd
,
τ= (τ1, ..., τd)
and
(Bn:= B(xn, rn))n∈N
be a sequence of balls of
Rd
satisfying
rn→0
and
µ(lim supn→+∞Bn) = 1.
Dene
Rn=˚
Rτ(xn, rn),
where
Rτ(xn, rn) = xn+
d
Y
i=1
[−1
2rτi,1
2rτi].
(7.56)
Then
dimH(lim sup
n→+∞
Rn)≥min
1≤i≤d®dim(µ) + P1≤j≤iτi−τj
τi´.
(7.57)
Remark 7.3.7.
(1) Since
(τ1, ..., τd)7→ min1≤i≤dndim(µ)+P1≤j≤iτi−τj
τio
is con-
tinuous, the result stands for the sequence of closed rectangles as well.

7.3 UBIQUITY THEOREMS AND SELF-CONFORMAL MEASURES139
(2) One may also apply any rotation to the shrunk rectangles, this wouldn't
change the bound (since Hausdor contents are invariant by rotation).
(3) As said above, when
K
is the closure of its interior is that it is easy
to compute
Hs
∞(Rn∩K).
Without this assumption, the conclusion of Theorem
7.3.6 fails. Indeed, in general, no formula involving only the dimension of
the measure and the contraction ratio can be accurate. For instance, consider
a self-similar measure in
R2
carried by a line
D
and a sequence
(Bn)n∈N
of
balls centered on the attractor
K
and verifying
µ(lim supn→+∞Bn) = 1.
Then,
consider the sequence of rectangles
Rn
with side-length
|Bn|τ1×|Bn|τ2
,
1≤τ1≤
τ2
and where the largest side (of side-length
|Bn|τ1
) is in the direction of
D
.
In this case, Theorem 7.3.1 yields the lower-bound
dimH(lim supn→+∞Rn)≥
dimH(µ)
τ1
. Then if
b
Rn
are the rectangles
Rn
rotated by
π
2
, Theorem 7.3.1 gives
that
dimH(lim supn→+∞b
Rn)≥dimH(µ)
τ2
. Moreover, under additional conditions,
these lower bounds are equalities.
Proof.
Given
τ1= 1 ≤τ2≤... ≤τd
and
s≥0
, set
τ= (τ1, . . . , τd)
and
gτ(s) = max
1≤k≤d(sτk−X
1≤i≤k
τk−τi).
We will need the following lemma (one refers to [51], Proposition 2.1 for
the proof, although it is stated in terms of singular values functions).
Lemma 7.3.8.
Let
τ1= 1 ≤τ2≤... ≤τd
.
The are two positive constants
C1
and
C2
depending on
d
only such that
for all
s≥0
,
r > 0
and
x∈Rd
one has
C1rgτ(s)≤ Hs
∞(Rτ(x, r)) = Hs
∞(˚
Rτ(x, r)) ≤C2rgτ(s).
Recall that
K
is the closure of its interior, and note that since the weights
pi
are taken positive in Denition 4.3.1, one must have
µ(˚
K)>0.
Denote
eµ=µ(·)
µ(˚
K)
and
α= dim(µ) = dim(eµ).
It is easily veried that the
computation made in the proof of Theorem 6.2.4 implies that, for any,
ε′>0
and any open set
Ω⊂˚
K
, there exists a constant
c(d, µ, s, ε′)
given by Theorem
6.2.4, so that



c(µ, d, s, ε′)Hs+ε′
∞(Ω) ≤ H˜µ,s
∞(Ω) ≤ Hs
∞(Ω)
if
s<α
H˜µ,s
∞(Ω) = 0
if
s > α.
(7.58)
Also,
eµ
being absolutely continuous with respect to
µ
, the sequence
(Bn)