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Self-similar sets and measures on the line
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Abstract
We discuss the problem of determining the dimension of self-similar sets and measures on . We focus on the developments of the last four years. At the end of the paper, we survey recent results about other aspects of self-similar measures including their Fourier decay and absolute continuity.
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We show that for Lebesgue almost all d -tuples (\theta_1,\ldots,\theta_d) , with |\theta_j|>1 , any self-affine measure for a homogeneous non-degenerate iterated function system \{Ax+a_j\}_{j=1}^m in \mathbb{R}^d , where A^{-1} is a diagonal matrix with the entries (\theta_1,\ldots,\theta_d) , has power Fourier decay at infinity.
Baker (2019), Bárány and Käenmäki (2019) independently showed that there exist iterated function systems without exact overlaps and there are super-exponentially close cylinders at all small levels. We adapt the method of Baker and obtain further examples of this type. We prove that for any algebraic number there exist real numbers such that the iterated function system satisfies the above property.
We introduce a parameter space containing all algebraic integers β∈(1,2] that are not Pisot or Salem numbers, and a sequence of increasing piecewise continuous function on this parameter space which gives a lower bound for the Garsia entropy of the Bernoulli convolution νβ. This allows us to show that dimH(νβ)=1 for all β with representations in certain open regions of the parameter space.
Let be a Borel probability measure on with a finite exponential moment, and assume that the subgroup generated by the support of is Zariski dense. Let be the unique stationary measure on . We prove that the Fourier coefficients of converge to 0 as tends to infinity. Our proof relies on a generalized renewal theorem for the Cartan projection.
In this note we present some one-parameter families of homogeneous self-similar measures on the line such that - the similarity dimension is greater than 1 for all parameters and - the singularity of some of the self-similar measures from this family is not caused by exact overlaps between the cylinders. We can obtain such a family as the angle- projections of the natural measure of the Sierpi\'nski carpet. We present more general one-parameter families of self-similar measures , such that the set of parameters for which is singular is a dense set but this "exceptional" set of parameters of singularity has zero Hausdorff dimension.
Denote by μa the distribution of the random sum (1 - a), where P(ωj = 0) = P(ωj = 1) = 1/2 and all the choices are independent. For 0 < a < 1/2, the measure μa is supported on Ca, the central Cantor set obtained by starting with the closed united interval, removing an open central interval of length (1 - 2a), and iterating this process inductively on each of the remaining intervals. We investigate the convolutions μa* (μb {ring operator} Sλ-1), where Sλ(x) = λx is a rescaling map. We prove that if the ratio log b/ log a is irrational and λ ≠ 0, then, where D denotes any of correlation, Hausdorff or packing dimension of a measure. We also show that, perhaps surprisingly, for uncountably many values of λ the convolution μ1/4* (μ1/3 {ring operator} Sλ-1) is a singular measure, although dimH(C1/4) + dimH(C1/3) > 1 and log(1/3)/ log(1/4) is irrational.
Let {F} be a self-similar set on \mathbb{R} associated to contractions {f_j(x) = r_j x + b_j} , {j \in \mathcal{A}} , for some finite \mathcal{A} , such that {F} is not a singleton. We prove that if {\log r_i / {\log r_j}} is irrational for some {i \neq j} , then {F} is a set of multiplicity, that is, trigonometric series are not in general unique in the complement of {F} . No separation conditions are assumed on {F} . We establish our result by showing that every self-similar measure {\mu} on {F} is a Rajchman measure: the Fourier transform {\widehat{\mu}(\xi) \to 0} as {|\xi| \to \infty} . The rate of {\widehat{\mu}(\xi) \to 0} is also shown to be logarithmic if {\log r_i / {\log r_j}} is diophantine for some {i \neq j} . The proof is based on quantitative renewal theorems for stopping times of random walks on \mathbb{R} .
We establish a complete algebraic characterization of self-similar iterated function systems Φ on Rd, for which there exists a positive probability vector p so that the Fourier transform of the self-similar measure corresponding to Φ and p does not tend to 0 at infinity.
We consider one-parameter families of smooth uniformly contractive iterated function systems {fjλ} on the real line. Given a family of parameter dependent measures {μλ} on the symbolic space, we study geometric and dimensional properties of their images under the natural projection maps Πλ. The main novelty of our work is that the measures μλ depend on the parameter, whereas up till now it has been usually assumed that the measure on the symbolic space is fixed and the parameter dependence comes only from the natural projection. This is especially the case in the question of absolute continuity of the projected measure (Πλ)⁎μλ, where we had to develop a new approach in place of earlier attempt which contains an error. Our main result states that if μλ are Gibbs measures for a family of Hölder continuous potentials ϕλ, with Hölder continuous dependence on λ and {Πλ} satisfy the transversality condition, then the projected measure (Πλ)⁎μλ is absolutely continuous for Lebesgue a.e. λ, such that the ratio of entropy over the Lyapunov exponent is strictly greater than 1. We deduce it from a more general almost sure lower bound on the Sobolev dimension for families of measures with regular enough dependence on the parameter. Under less restrictive assumptions, we also obtain an almost sure formula for the Hausdorff dimension. As applications of our results, we study stationary measures for iterated function systems with place-dependent probabilities (place-dependent Bernoulli convolutions and the Blackwell measure for binary channel) and equilibrium measures for hyperbolic IFS with overlaps (in particular: natural measures for non-homogeneous self-similar IFS and certain systems corresponding to random continued fractions).
In this note we present an algorithm to obtain a uniform lower bound on Hausdorff dimension of the stationary measure of an affine iterated function scheme with similarities, the best known example of which is Bernoulli convolution. The Bernoulli convolution measure μλ is the probability measure corresponding to the law of the random variableξ=∑k=0∞ξkλk, where ξk are i.i.d. random variables assuming values −1 and 1 with equal probability and 12<λ<1. In particular, for Bernoulli convolutions we give a uniform lower bound dimH(μλ)≥0.96399 for all 12<λ<1.
Let Φ be a C1+γ smooth IFS on R, where γ>0. We provide mild conditions on the derivative cocycle that ensure that every self conformal measure is supported on points x that are absolutely normal. That is, for every integer p≥2 the sequence {pkx}k∈N equidistributes modulo 1. We thus extend several state of the art results of Hochman and Shmerkin [29] about the prevalence of normal numbers in fractals. When Φ is self-similar we show that the set of absolutely normal numbers has full Hausdorff dimension in its attractor, unless Φ has an explicit structure that is associated with some integer n≥2. These conditions on the derivative cocycle are also shown to imply that every self conformal measure is a Rajchman measure, that is, its Fourier transform decays to 0 at infinity. When Φ is self similar and satisfies a certain Diophantine condition, we establish a logarithmic rate of decay.
We prove that, after removing a zero Hausdorff dimension exceptional set of parameters, all self-similar measures on the line have a power decay of the Fourier transform at infinity. In the homogeneous case, when all contraction ratios are equal, this is essentially due to Erdős and Kahane. In the non-homogeneous case the difficulty we have to overcome is the apparent lack of convolution structure.
We exhibit self-similar sets on the line which are not exponentially separated and do not generate any exact overlaps. Our result shows that the exponential separation, introduced by Hochman in his groundbreaking theorem on the dimension of self-similar sets, is too weak to describe the full theory.
Several important conjectures in Fractal Geometry can be summarised as follows: If the dimension of a self-similar measure in R does not equal its expected value, then the underlying iterated function system contains an exact overlap. In recent years significant progress has been made towards these conjectures. Hochman proved that if the Hausdorff dimension of a self-similar measure in R does not equal its expected value, then there are cylinders which are super-exponentially close at all small scales. Several years later, Shmerkin proved an analogous statement for the Lq dimension of self-similar measures in R. With these statements in mind, it is natural to wonder whether there exist iterated function systems that do not contain exact overlaps, yet there are cylinders which are super-exponentially close at all small scales. In this paper we show that such iterated function systems do exist. In fact we prove much more. We prove that for any sequence (ϵn)n=1∞ of positive real numbers, there exists an iterated function system {ϕi}i∈I that does not contain exact overlaps andmin{|ϕa(0)−ϕb(0)|:a,b∈In,a≠b,ra=rb}≤ϵn for all n∈N.
We present a self-contained proof of a formula for the dimensions of self-similar measures on the real line under exponential separation (up to the proof of an inverse theorem for the norm of convolutions). This is a special case of a more general result of the author from Shmerkin (Ann Math, 2019), and one of the goals of this survey is to present the ideas in a simpler, but important, setting. We also review some applications of the main result to the study of Bernoulli convolutions and intersections of self-similar Cantor sets.
The Bernoulli convolution νλ with parameter λ ϵ (0, 1) is the probability measure supported on R that is the law of the random variable ∑±λn, where the ± are independent fair coin-tosses. We prove that dim νλ = 1 for all transcendental λ ϵ (1/2, 1).
We give an expression for the Garsia entropy of Bernoulli convolutions in terms of products of matrices. This gives an explicit rate of convergence of the Garsia entropy and shows that one can calculate the Hausdorff dimension of the Bernoulli convolution to arbitrary given accuracy whenever is algebraic. In particular, if the Garsia entropy is not equal to then we have a finite time algorithm to determine whether or not .
This paper is concerned with the Diophantine properties of the sequence{ξθⁿ}, where1 ≤ ξ < θ and θ is a rational or an algebraic integer. We establish a combinatorial proposition which can be used to study such two cases in the same manner. It is shown that the decay rate of the Fourier transforms of self-similar measuresμλ withλ = θ⁻¹ as the uniform contractive ratio is logarithmic. This generalizes some results of Kershner and Bufetov-Solomyak, who consider the case of Bernoulli convolutions. As an application, we prove thatμλ almost every x is normal to any base b ≥ 2, which implies that there exist infinitely many absolute normal numbers on the corresponding self-similar set. This can be seen as a complementary result of the well-known Cassels-Schmidt theorem.
We prove that self-similar measures on the real line are absolutely continuous for almost all parameters in the super-critical region, in particular confirming a conjecture of S-M. Ngai and Y. Wang. While recently there has been much progress in understanding absolute continuity for homogeneous self-similar measures, this is the first improvement over the classical transversality method in the general (non-homogeneous) case. In the course of the proof, we establish new results on the dimension and Fourier decay of a class of random self-similar measures.
The Bernoulli convolution with parameter is the probability measure that is the law of the random variable , where the signs are independent unbiased coin tosses. We prove that each parameter with can be approximated by algebraic parameters within an error of order for any number A, such that . As a corollary, we conclude that for each of . These are the first explicit examples of such transcendental parameters. Moreover, we show that Lehmer's conjecture implies the existence of a constant such that for all .
We study a class of measures on the real line with a kind of self-similar structure, which we call dynamically driven self-similar measures, and contain proper self-similar measures such as Bernoulli convolutions as special cases. Our main result gives an expression for the -dimensions of such dynamically driven self-similar measures, under certain conditions. As an application, we settle Furstenberg's long-standing conjecture on the dimension of the intersections of and -invariant sets. Among several other applications, we also show that Bernoulli convolutions have an density for all finite q, outside of a zero-dimensional set of exceptions. The proof of the main result is inspired by M. Hochman's approach to the dimensions of self-similar measures and his inverse theorem for entropy. Our method can be seen as an extension of Hochman's theory from entropy to norms, and likewise relies on an inverse theorem for the decay of norms of discrete measures under convolution. This central piece of our approach may be of independent interest, and is an application of well-known methods and results in additive combinatorics: the asymmetric version of the Balog-Szemer\'{e}di-Gowers Theorem due to Tao-Vu, and some constructions of Bourgain.
We prove that Bernoulli convolutions are absolutely continuous provided the parameter lambda is an algebraic number sufficiently close to 1 depending on the Mahler measure of lambda.
Absolute continuity and singularity play a very important role in the study of measures in infinite-dimensional spaces, for example, Hubert space. Although there can be no theory treating such questions for finite-dimensional spaces which of great interest, such a theory for infinite-dimensional spaces is possible. It contains such topics as the investigation of the absolute continuity and singularity of various concrete classes of measures, the finding of general conditions for absolute continuity or singularity in terms of finite-dimensional distributions, and other characteristics defining the measures. An important problem is the calculation of the density of a measure w.r.t. another when the measures are absolutely continuous and the determination of the non-overlapping sets on which singular measures are concentrated.
The exponential growth rate of non polynomially growing subgroups of
is conjectured to admit a uniform lower bound. This is known for non-amenable
subgroups, while for amenable subgroups it is known to imply the Lehmer
conjecture from number theory. In this note, we show that it is equivalent to
the Lehmer conjecture. This is done by establishing a lower bound for the
entropy of the random walk on the semigroup generated by the maps , where is an algebraic number. We give a bound
in terms of the Mahler measure of . We also derive a bound on the
dimension of Bernoulli convolutions.
We study products of random isometries acting on Euclidean space. Building on
previous work of the second author, we prove a local limit theorem for balls of
shrinking radius with exponential speed under the assumption that a Markov
operator associated to the rotation component of the isometries has spectral
gap. We also prove that certain self-similar measures are absolutely continuous
with smooth densities. These families of self-similar measures give higher
dimensional analogues of Bernoulli convolutions on which absolute continuity
can be established for contraction ratios in an open set.
We prove that the set of exceptional such that the
associated Bernoulli convolution is singular has zero Hausdorff dimension, and
likewise for biased Bernoulli convolutions, with the exceptional set
independent of the bias. This improves previous results by Erd\"os, Kahane,
Solomyak, Peres and Schlag, and Hochman. A theorem of this kind is also
obtained for convolutions of homogeneous self-similar measures. The proofs are
very short, and rely on old and new results on the dimensions of self-similar
measures and their convolutions, and the decay of their Fourier transform.
The paper gives first quantitative estimates on the modulus of continuity of
the spectral measure for weakly mixing suspension flows over substitution
automorphisms. The main results are, first, a Hoelder estimate for the spectral
measure of almost all suspension flows with a piecewise constant roof function;
second, a log-Hoelder estimate for self-similar suspension flows; and, third, a
Hoelder asymptotic expansion of the spectral measure at zero for such flows.
The second result implies log-Hoelder estimates for the spectral measures of
translation flows along stable foliations of pseudo-Anosov automorphisms. The
Appendix explains the connection of these results with the theory of Bernoulli
convolutions.
We study the Hausdorff dimension of self-similar sets and measures on the
line. We show that if the dimension is smaller than the minimum of 1 and the
similarity dimension, then at small scales there are super-exponentially close
cylinders. This is a step towards the folklore conjecture that such a drop in
dimension is explained only by exact overlaps, and confirms the conjecture in
cases where the contraction parameters are algebraic. It also gives an
affirmative answer to a conjecture of Furstenberg, showing that the projections
of the "1-dimensional Sierpinski gasket" in irrational directions are all of
dimension 1.
As another consequence, if a family of self-similar sets or measures is
parametrized in a real-analytic manner, then, under an extremely mild
non-degeneracy condition, the set of "exceptional" parameters has Hausdorff
dimension 0. Thus, for example, there is at most a zero-dimensional set of
parameters 1/2<r<1 such that the corresponding Bernoulli convolution has
dimension <1, and similarly for Sinai's problem on iterated function systems
that contract on average.
A central ingredient of the proof is an inverse theorem for the growth of
Shannon entropy of convolutions of probability measures. For the dyadic
partition D_n of the line into intervals of length 1/2^n, we show that if
H(nu*mu,D_n)/n < H(mu,D_n)/n + delta for small delta and large n, then, when
restricted to random element of a partition D_i, 0<i<n, either mu is close to
uniform or nu is close to atomic. This should be compared to results in
additive combinatorics that give the global structure of measures satisfying
H(nu*mu,D_n)/n < H(mu,D_n)/n + O(1/n).
Refinable functions and distributions with integer dilations have been studied extensively since the pioneer work of Daubechies on wavelets. However, very little is known about refinable functions and distributions with non-integer dilations, particularly concerning its regularity. In this paper we study the decay of the Fourier transform of refinable functions and distributions. We prove that uniform decay can be achieved for any dilation. This leads to the existence of refinable functions that can be made arbitrarily smooth for any given dilation factor. We exploit the connection between algebraic properties of dilation factors and the regularity of refinable functions and distributions. Our work can be viewed as a continuation of the work of Erdös [P. Erdös, On the smoothness properties of a family of Bernoulli convolutions, Amer. J. Math. 62 (1940) 180–186], Kahane [J.-P. Kahane, Sur la distribution de certaines séries aléatoires, in: Colloque de Théorie des Nombres, Univ. Bordeaux, Bordeaux, 1969, Mém. Soc. Math. France 25 (1971) 119–122 (in French)] and Solomyak [B. Solomyak, On the random series ∑±λn (an Erdös problem), Ann. of Math. (2) 142 (1995) 611–625] on Bernoulli convolutions. We also construct explicitly a class of refinable functions whose dilation factors are certain algebraic numbers, and whose Fourier transforms have uniform decay. This extends a classical result of Garsia [A.M. Garsia, Arithmetic properties of Bernoulli convolutions, Trans. Amer. Math. Soc. 102 (1962) 409–432].
Let be an iterated function system (IFS) on with attractor K. Let denote the one-sided full shift over the alphabet . We define the projection entropy function on the space of invariant measures on associated with the coding map , and develop some basic ergodic properties about it. This concept turns out to be crucial in the study of dimensional properties of invariant measures on K. We show that for any conformal IFS (resp., the direct product of finitely many conformal IFS), without any separation condition, the projection of an ergodic measure under is always exactly dimensional and, its Hausdorff dimension can be represented as the ratio of its projection entropy to its Lyapunov exponent (resp., the linear combination of projection entropies associated with several coding maps). Furthermore, for any conformal IFS and certain affine IFS, we prove a variational principle between the Hausdorff dimension of the attractors and that of projections of ergodic measures. Comment: 60 pages
In this paper we study the absolute continuity of self-similar measures defined by iterated function systems (IFS) whose contraction ratios are not uniform. We introduce a transversality condition for a multi-parameter family of IFS and study the absolute continuity of the corresponding self-similar measures. Our study is a natural extension of the study of Bernoulli convolutions by Solomyak, Peres, et al.
On self-similar sets with overlaps and inverse theorems for entropy in R d
- M Hochamn
M. Hochamn, On self-similar sets with overlaps and inverse theorems for
entropy in R d, 2017. To appear in Memoirs of the Amer. Math. Soc.
arXiv:1503.09043v2.