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The Problem of the Infinite Exponential

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Same description as that for the Greek version of the thesis, except in English for help with International Researchers. Cantor Bouquets are exponentially powered Cracks that move unpredictably on the Hilbert Kernel. Analytic profil of the topology in the Complex Plane of the iterated exponential map, with companion Maple code which shows Fatou and Julia domains.
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The Problem of the Infinite Exponential
Ioannis N. Galidakis,
Mathematician,
Agricultural University,
Athens, Greece
August 2020
Dissertation for the title of “Doctor of Mathematics” at the Agricultural University
of Athens, submitted by Ioannis Galidakis, Mathematician from Athens, in August 30,
2020.
1
Advisor: Professor Ioannis Papadoperakis
Co-advisors: Professors Dimitrios Gatzouras, Charalambos Charitos
2
To the memory of my Parents
3
Contents
1 Introduction 5
2 An application of function Wto infinite exponentials 6
2.1 Denitions.................................... 6
2.2 TheWfunction................................. 7
2.3 Thecentrallemma ............................... 10
2.4 Convergence for xR............................. 11
2.5 Convergence for cC............................. 14
2.6 Convergence for qQ............................. 15
3 An application of functions HW to infinite exponentials 16
3.1 Denitions.................................... 16
3.2 The central lemma on HW . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4 Topology of the infinite exponential 18
4.1 JuliaandFatousets .............................. 18
4.2 Attractors and repellers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.3 General topological map of the infinite exponential . . . . . . . . . . . . . 20
4.4 Topology .................................... 22
4.5 Fractal nature in the infinite exponential . . . . . . . . . . . . . . . . . . . 22
4.6 The primitive of a Cantor bouquet . . . . . . . . . . . . . . . . . . . . . . 26
4.7 Pinball with the infinite exponential . . . . . . . . . . . . . . . . . . . . . 27
5 The generalized infinite exponential 27
5.1 Denitions.................................... 27
5.2 Ascendingindexes ............................... 28
5.3 Descendingindexes............................... 29
6 Appendix: programming/results 31
7 Acknowledgements 33
8 Bibliography 34
4
1 Introduction
The problem of the infinite exponential was observed and studied for the first time by
Leonhard Euler ([85]). It concerns the sequence defined recursively as follows:
Definition 1.1 Given x > 0,
αn+1 =xαn
The central problem of the infinite exponential is, then, the Proposition:
Proposition 1.2 Given x > 0, study the behavior of the recursive sequence αn,nN.
Euler proved the following theorem:
Theorem 1.3 The sequence αnconverges if and only if xee, ee1.
The central problem leads to an extensive labyrinth of sub-problems, which are ex-
amined partially or fully in the bibliography references. The reason why the area of
sub-problems is deep and extensive, is that the sequence (1.1) is based on the arith-
metic operator of exponentiation, which is the most “powerful” of the three {+,×,↑},
but unfortunately it is the only one that is not commutative. In other words, in general1,
ab̸=ba.
A second reason the central problem leads to a plethora of sub-problems, is that
the two previous operators do not exhibit any idiopathic behavior relative to their re-
peated application. For example, for given x > 1, the repeated operations +(n)x=
x+x+...+x
  
n
and ×(n)x=x×x×. . . ×x
  
n
, cause divergence. The latter does not al-
ways happen with the exponentiation operator, as the sequence (1.1), may converge as
with Euler’s Theorem, for xee, ee1.
The above difference becomes immediately obvious for example, when a student at-
tempts to investigate the behavior of the sequence (1.1), for given x > 1, let’s say for
x= 1.2, where a simple pocket calculator shows that the repeated exponentiation of such
xleads to an unexpected convergence.
This unexpected convergence is the exceptionally strange of theorem (1.3), something
which was observed by Euler, using only paper and pencil. Something like that is really
inconceivable without the use of computers, at least insofar modern calculation meth-
ods are concerned, as the exponentiation operator exhibits mainly calculation difficulties
which do not exist when using the other two previous operators, since repeated addition
and multiplication are relatively “fast” operations when using computers.
We, then have a problem which could possibly be investigated further using fairly
strong computer systems, which could conceivably help with many of the sub-problems
that show up.
The first important development came around 1999-2000, when fairly advanced ver-
sions of the Computer Algebra Systems (CAS) Maple and Mathematica showed up [154].
1The problem of solving the equation ab̸=ba, which is thus of substantial importance in the inves-
tigation of the infinite exponential, was presented for the first time to the author in 1982, in a letter of
the author’s Father to the author
5
These packages include since then symbolic versions of the complex function W, otherwise
known as Lambert’s function. This latter function has found multiple applications the
last ten to fifteen years in various sectors of pure and applied mathematics, like physics,
electronics, optics and astronomy.
The function W is a limited inverse of the complex function z·ez, with the latter
being intrinsically connected to the operator of the repeated exponentiation, because the
last one can be analyzed in parts by z·ez, when one considers the partial sub-exponents
of a repeated exponential, using the primitive base exp.
Finally and after years of research in arithmetical analysis of the results of the survey
related to the infinite exponential and the Maple CAS Maple ([67, 332]), it is validated
that the function W, gives a closed form for the convergence limit of the sequence (1.1)
as it follows from Theorem (1.3), when the sequence (1.1) converges.
Therefore the function W becomes the ideal tool to examine the infinite exponential
with and consequently allows a direct extraction of final conclusions related to the area
of convergence of the sequence (1.1), extending Theorem (1.3) to the complex plane, to
an area called today area Shell-Thron.
The above results are generalized directly to the space of Quaternions, with the new
results coinciding with the already known ones for the complex plane, despite the Quater-
nion space not being (in general) commutative.
The problem of the infinite exponential does not end there, as arithmetic calculations
also suggest non-convergence or even divergence of the sequence in many sub-areas of the
complex plane, which need additional calculations in order to find out if in these areas
the sequence (1.1) behaves normally and how.
Examining these areas of unknown behavior of the infinite exponential, requires the
use of functions which are gradually more complex in nature than W and the author calls
them HW. These functions solve additionally the cases where the sequence (1.1) falls
into cycles of period p > 1, characterizing therefore fully the sub-areas of convergence of
the sequence, when there are multiple attractors, which are given in closed form by the
functions HW, and determine therefore the partial limits of the convergence.
This work collects sub-results until Proposition (1.2) of Euler is answered in its general
form, which is essentially the topology of the sequence αn+1 =γαn
n, for general γnC,
calculates a dimension bound for its main feature set and examines partially the behavior
of the functions HW which allow the extraction of conclusions for this Proposition.
2 An application of function Wto infinite exponentials
2.1 Definitions
The function W acquired substantial publicity lately, mainly because of important progress
in computational mathematics. Even though compositions of it appear in hidden form
in references [18, 153], [187, 14] and [122, 235], its essential properties are presented in
[67, 344-349] and [68, 2-8]. Some of these properties can be used to simplify the answer
of when the infinite exponential converges.
We are working with the principal branch of the complex map log, and we use Maurer’s
notation for repeated and infinite exponential (see [122, 239-240]).
6
It is assumed that equations with complex exponents anywhere in this dissertation
use always the principal branch of the complex exponentiation, whenever this is needed:
cw=ew·log(c),c̸= 0, with log always denoting the principal branch of the complex
logarithm log(k, z), for θ(π, π].
Definition 2.1 For zC\ {xR:x0}and nN,
nz=z, if n= 1,
z(n1z), if n > 1.
Whenever the following limit exists and is finite, we set:
z= lim
n→∞
nz(1)
We use the exponential map gc(z) and its compositions, for cC\ {xR:x0}.
Whenever the parameter cis obvious, it will be omitted to avoid any confusion.
gc(z) = cz(2)
g(n)
c(z) = gc(z) if n= 1,
gcg(n1)
c(z)if n > 1.(3)
nzand g(n)(z) are related: nc=g(n)
c(1). We also use the following function, which is
a partial inverse of W:
m(z) = z·ez, z C(4)
The terminology Infinite Exponential seems to have been used first in [18, 150]. In
short, it is the infinite tower zzz···
3
2
1, with znR(or znC), nN. We are initially in-
terested in the case zn=z,nN. Respectively, the terminology zz···z
(real or complex)
will be used alternatively to to the notation z(z(···z))to notate repeated exponentiation
from the top to the bottom. In the majority of the cases we use Definition (2.1) for the
repeated exponential.
2.2 The Wfunction
The complex function W is a partial inverse of m(z) or otherwise the function which
solves the equation m(z) = wrelative to z. Alternatively,
Definition 2.2 Wsatisfies the functional equation:
W(z)eW(z)=z, z C
7
Figure 1: Quadratrix of Hippias bounds the ranges of W(k, z), kZ
W is multivalued and has infinitely many branches as a complex Riemann surface. It
is usually notated as W(k, z), with kZspecifying the working branch. Specifically, its
principal branch W(0, z) corresponds to k= 0, and then the function is notated as W(z).
Follow some useful properties of W. Most of these can be found in [67] and [68] and
can be validated numerically using the Maple CAS in [154, 305]. We selectively prove
only some of the properties which are not immediately obvious in [68] or [67].
For kZ, the various branches of W(k, z ) are defined in Cand are discontinuous at
the points of the intervals BCk:
BCk=
−∞,e1, if k= 0
−∞,e1e1,0, if k=1
(−∞,0) , otherwise
We observe that the branch point z0=e1of W(z) is m(z), with zsatisfying:
dm
dz = 0. This branch point is shared between the two branches W(z) and W(1, z).
Suppose then that CN = (−∞,1) and let us consider the curves:
Ck=ycot(y) + yi, y (2, (2k+ 1)π), if k0,
ycot(y) + yi, y ((2k+ 1)π, (2k+ 2)π), if k < 0.
Lemma 2.3 The image of BCkunder W(k, z )is:
W(k, B Ck) = C1CN, if k=1
Ck, otherwise
We now define the regions Dkas follows:
Dk=
area between C1, CN , C0, if k= 1
area between C1, CN , C2, if k=1
area between Ck, Ck1, otherwise
8
The bounding curves of the ranges of W(k, z)Ckand the regions Dkgive the well
known quadratrix of Hippias, which is presented in [68] and in Figure 1.
The range of the images of W is then constrained by the regions Dk(see [67, 13-23]),
consequently,
Lemma 2.4 W(k , C\ {0}) = DkCk,kZ\ {0}, and W(0,C) = D0C0
W(k, z) is a partial inverse of m(z) in the aforementioned regions DkCk, consequently
(see [67])
Lemma 2.5 W(k , m(z)) = z,kZ,zDkand m(W(k, z)) = z,kZ,zC
Corollary 2.6 W(m(z)) = z,zD0and m(W(z)) = z,zC
From the symmetry between Ckand Dkfollows:
Lemma 2.7 W(k , z) = W(k, z),kZ,zC
Corollary 2.8 W(z) = W(z),zC
Since (−∞,1] D1and [1,+)D0, only the branches that correspond to
k= 0 and k=1 can ever assume real values.
Lemma 2.9 W(k , z)Rk∈ {−1,0}
From the first diagram in [67] (or elementary calculus), follow,
Lemma 2.10 W(x)is real, continuous and monotone increasing on the interval [e1,+).
Lemma 2.11 W(1, x)is real, continuous and monotone decreasing on the interval
[e1,0).
Lemmas (2.10) and (2.11) follow easily by considering the function m(z) and the fact
that e1is a common branch point between W(z) and W(1, z):
Lemma 2.12 W(e) = 1
Lemma 2.13 We1= W 1,e1=1
Lemma 2.14 DD0and D ∂D0={−1}
Proof: It suffices to show |−ycot(y) + yi|>1 for each y0,π
2. We observe
immediately that lim
y0+(ycot(y) + yi) = 1D ∂D0and zD ∂D0
cos(πy) + sin(πy)i=ycot(y) + yi ⇒ {sin(y) = y, ycot(y) = cos(πy)}. From
the first equation we get y=,kZ. From it only the equation y= (2k+ 1)π,kZ
satisfies the second equation as a limit, therefore z= cos((2k+1)π)+ sin((2k+1)π)i=1
and the Lemma follows.
9
Lemma 2.15 W(z)is analytic at the origin z0= 0 with series:
S(z) =
n=1
(n)n1zn
n!
and convergence radius: Rs=e1
Proof: Details related to the series S(z) as well as other expansions are given in [67].
The Ratio test reveals the radius of convergence. lim
n→∞
an+1
an= lim
n→∞ 1 + 1
nn1z=
|ez|<1, or equivalently, |z|< e1.S(z) is valid on the entire disk Dw=z:|z| ≤ e1.
When |z|=e1, then
(n)n1zn
n!=nn1
enn!<nn1
2πn(n+1
2)=1
2πn 3
2, using Stirling’s ap-
proximation and the series
n=1
1
2πn 3
2converges.
The fact that S(z) has radius of convergence Rs=e1, follows also from the fact that
W(z) has a branch point at z0=e1.
2.3 The central lemma
the main result of Lemma (2.18) which concerns the limit of the Euler sequence is men-
tioned in [67, 332]. The author thinks it deserves a deeper analysis, particularly after the
fact that W is a multivalued function.
g(z) of (2) is intricately connected with the infinite exponential. Generally speaking,
given cC,c /∈ {0,1}, if the sequence g(n)(z), nNconverges, then it must converge
to a fixed point of g(z) or equivalently the limit must satisfy the first auxiliary equation,
z=g(z) (5)
Equation (5) can always be solved through W.
Lemma 2.16 The fixed points of g(z)are given by h:Z×C7→ C, with:
h(k, c) = W(k, log(c))
log(c),kZ
Proof: z=g(z)z=czzezlog(c)= 1 ⇔ −zlog(c)ezlog(c)=log(c)
m(zlog(c)) = log(c)⇔ −zlog(c) = W(k, log(c)), kZ, from the Definition of
m(z), z=W(k,log(c))
log(c),kZand the Lemma follows.
Lemma 2.17 If cC\ee1, then h(k, c)is a repeller of g(z).
Proof: If c̸=ee1, then g(h(k , c)) = log(c)W(k,log(c))
log(c)=W(k, log(c)) ∈ −Dk,
from Lemma (2.4), therefore if k̸= 0, then |g(h(k , c))|>1, from Lemma (2.12), and the
Lemma follows.
If c=ee1, then g(h(1, c) = W(1,e1) = 1 (Lemma (2.11)).
The supposition c̸=ee1is important. Otherwise, g(h(k, c)) = Wk, e1= 1,
for k=1 (Lemma (2.17)).
Lemmas (2.16) and (2.17)lead to the central Lemma of the dissertation:
10
Lemma 2.18 (Corless) Whenever the limit of g(n)(z)exists finitely, its value is given
by:
h(c) = h(0, c) = W(log(c))
log(c)
therefore
lim
n→∞ g(n)(z) = lim
n→∞ g(n)(1) = c=h(c)
2.4 Convergence for xR
The fact that the repeated iterated exponential with base 2 converges to 2 can be
validated arithmetically in [70, 70] and [117, 66] and is explained analytically in [171,
434], [127, 77] and [139, 643-646]. Considering the following relation for the positive
square root of 2, then (2)2= 2. replacing the exponent of the relation with the left side
of the equation, we get the sequence g(n)(2), nN. From the previous Lemma when the
limit exists, limn→∞ g(n)(z) = 2, nNfor suitable z, consequently,
(2) = 2 (6)
This will also be shown with the last Lemma of this section.
Lemma 2.19 If xe1, e, then hxx1=x.
Proof: By definition, h(y) solves the equation x=yxwhich is equivalent to xx1=y,
therefore it is a partial inverse of y(x) = xx1. In the given interval y(x) is 1-1 and onto
the range e1, e, and the Lemma follows.
Lemma 2.20 If x(e, +), then hxx1=w(1, e), with ww1=xx1.
Proof: y(x) is continuous on (1,+), acquires a maximum at x=e, and lim
x→∞ y(x) = 1,
therefore there exists unique win (1, e), such that ww1=xx1, and the Lemma follows
from lemma (2.19).
Example: y= 1.3304 .
= 1.562(1
1.562 ).
= 6.620(1
6.620 ). Such values are given in closed form
using W, but can also be given arithmetically or using other methods in the references
which deal with the solution of the equation xy=yx, like [43, 763], [55, 222-226], [56,
78-83], [89, 137], [141, 233-237], [161, 316], [169, 444-447], [183, 141], and [125, 96-99]. In
the references the authors observe that y(x) is a partial inverse of h(x), but don’t define
hthrough W.
Lemmas (2.19) and (2.20) in short:
Lemma 2.21
hxx1=x, if xe1, e;
w, w (1, e) : ww1=xx1,if x(e, +).
11
Lemma 2.22 If c=ee,x0= log Wlog(c)1log(c)1log(c)1, and u(x) = g(2) (x)
x, then,
x0is the only critical point of u(x), in [0,1] (7a)
x0=e1(7b)
u(x0) = 0 (7c)
du
dxx0
= 0 (7d)
du
dx <0,x[0,1] − {x0}(7e)
Proof: du
dx = 0 can be solved exactly through W. If du
dx = 0, then g(2) (x)g(x) log(c)2= 1,
therefore eylog(c)ylog(c)= log(c)1, where y=cx. Then m(ylog(c)) = log(c)1, therefore
ylog(c)= W k, log(c)1, consequently y= W k , log(c)1log(c)1, and finally x=
log Wk, log(c)1log(c)1log(c)1,kZ. (7a) follows directly from Lemma (2.5).
(7b) follows from Lemma (2.11). (7c) and (7d) follow easily. For (7e) we observe that
log(c) = e < 0, therefore, if x < x0then g(x)> e1and g(2)(x)< e1, therefore
g(2)(x)g(x) log(c)2<1, consequently du
dx <0. For x > x0the proof (with inequality signs
reversed) is similar and the Lemma follows.
Lemma 2.23 If cee, ee1, then c=h(c).
Proof: If c=ee, the fixed point of g(x) is given by Lemma (2.18). h(c) = h(ee) =
W(log(ee))
log(ee)=W(e)
e=e1, based on Lemma (2.26). Using Lemma (2.22), the continuity
of u(x) and the factoids: u(0) = c > 0, u(1) = cc1<0, it follows that g(2) (x)> x, if
x0, e1and g(2)(x)< x, if xe1,1. Using the two inequalities and induction
on n, the sequence: an=g(n)(1), nNsatisfies, a2n+2 < a2n, and a2n+3 > a2n+1, for
each nN. The last shows that a2n+1 and a2nare monotone increasing and decreasing,
respectively. Additionally, since 0 < c =ee<1, both sequences are bounded above by
1 and below by 0. It follows that both sequences a2n+1 and a2npossess limits. Since the
only root of u(x) is x0(otherwise equation (7a) is violated), both sequences converge to
x0, from which follows that anconverges to x0=e1.
If c=ee1, then the fixed point of g(x) is given again by Lemma (2.18). h(c) =
hee1=W(log(ee1))
log(ee1)=W(e1)
e1=e, based on Lemma (2.27). Using induction on
n, the sequence an=g(n)(1), nNis monotone increasing and bounded above by e,
therefore it converges to eand the Lemma follows.
[122, 240], [18, 153] and [187, 14-15] arrive at the same result differently, without using
the function W.
Lemma 2.24 If c(0, ee), then cdoes not exist.
Proof: The fixed point h(c) of g(x) from Lemma (2.18) is a repeller. If c(0, ee), then
W(e)<W(log(c)) based on Lemma (2.10) and consequently 1 <W(log(c)), based
on Lemma (2.26). This means 1 <|g(h(c))|and the Lemma follows from fixed point
iteration.
12
Lemma 2.25 If cee, ee1, then c=h(c).
Proof: The fixed point h(c) of g(x) from Lemma (2.18) is an attractor. If cee, ee1,
then e1<log(c)< e, therefore W e1<W(log(c)) <W(e), based on Lemma
(2.7) and consequently |g(h(c))|<1 based on Lemmas (2.10) and (2.11) and the Lemma
follows from fixed point iteration.
Lemma 2.26 If cee1,+, then cdoes not exist.
Proof: The fixed point h(c) of g(x) from Lemma (2.18) is a repeller. If cee1,+,
then log(c)BC0, therefore W(log(c)) C0, based on Lemma (2.12), and conse-
quently |g(h(c))|>1, based on the same Lemma and the Lemma follows from fixed
point iteration.
Lemmas (2.23)-(2.26) and (2.18) validate the final Lemma of this section which is
Theorem (1.3) of Euler:
Lemma 2.27 cexists, then and only, when cee, ee1, me c=h(c).
Proof: The interval of convergence can be determined for the real case from fixed
point iteration. The only potential fixed point of g(x) is given from Lemma (2.18), as
h(c). Using elementary properties of all the relevant functions, if |g(h(c))| ≤ 1, then
|−W(log(c))| ≤ 1. This means W(log(c)) [1,1], therefore m(W(log(c)))
m([1,1]), or m(W(log(c))) e1, e, therefore also log(c)e1, e, based on
Definition (2.2) and finally cee, ee1.
Using Lemma (2.27) for c=ee1,c=eeand c=2,
ee1=ee1(ee1)···
=hee1=e
ee=ee(ee)···
=hee=e1
2=2(2)···
=h21
2= 2
The last equations settle normally the question in the beginning of this section with
equation (6). That the infinite exponential converges if and only iff its base belongs to
the interval ee, ee1.
= [0.06598,1.44466] is also shown in [122, 240], [139, 645] and
[146, 556] using other methods and without employing the function W.
[9, 207-208] and [135, 301-303] also show that for kN, lim
c0+
2kc= 1 and lim
c0+
2k+1c=
0. If c(0, ee), then nc,nNis a cycle of period 2 by considering the even and odd
subsequences. The bifurcation that occurs and its behavior are analyzed in [9, 207], [187,
15] and [135, 299]. We note that the two branches which stem from the bifurcation point
ee, e1can be given in parametric form as a(a
1a)and a(1
1a)for proper positive a
(see for example [122, 237] or [188, 212]). In this case, as is shown in [171, 434], [122,
13
241-243], [187, 13] and [129, 501], the two separate limits a= lim
n→∞
2n+1cand b= lim
n→∞
2nc
satisfy the inequality 0 < a < h(c)< b < 1 and the second auxiliary system,
a=cca
b=ca(8)
A closed form solution for system (8) is given in detail in a subsequent section, where
the difficulties of solving the auxiliary equation of order nare presented, for the complex
exponential map g(z), using certain generalized functions based on W.
2.5 Convergence for cC
Suppose Dis the unit disk. We consider the map ϕ:C7→ C, defined as: ϕ(z) = e(z
ez)=
em(z). The image RS T =ϕ(D) is a nephroid region called Shell-Thron region, with
an approximation given in [210] and in Figure 2.
Figure 2: Shell-Thron region
[165, 679], [164, 12] and [13, 106] show that on the interior of RST we have convergence
of g(n)(z). What happens at the boundary of the figure is mentioned in [12, 502] and [11]:
Theorem 2.28 (Baker/Rippon) The sequence nc,nNconverges for log(c)∈ {tet:
|t|<1, or tn= 1, for some nN}and diverges elsewhere.
tand care related through W, by considering always the principal branches of all
maps involved. c=ϕ(t)W (log(c)) = W (m(t)) t=W (log(c)), using
Corollary(2.6) and Lemma (2.12). Consequently ϕis reversible and ϕ1= (W)(log)
with t=ϕ1(c). Then we have the Theorem:
Theorem 2.29 The sequence nc,nNconverges if ϕ1(c)<1or ϕ1(c)n= 1,
where ϕ1=W(log(c)), some nNand converges elsewhere.
Lemma 2.30 If cC, then the multipler of the fixed point h(c)of g(z)is given by
t=ϕ1(c).
14
Proof:
g(h(c)) = log(c)·g(h(c))
= log(c)·h(c)
= log(c)·W (log(c))
log(c)
=W(log(c)) = t
and the Lemma follows.
Theorem (2.28) then alternatively states that when c=ϕ(t), then the sequence g(n)(c),
nNconverges if and only if the measure of the multiplier t=ϕ1(c) of the fixed point
h(c) of g(z) is less than 1, or the multiplier is a n-th root of unity.
Lemma 2.31 If |t|<1and c=ϕ(t), then c=h(c).
Proof: The fixed point of g(z) is given by Lemma (2.18), as h(c). h(c) = W(log(c))
log(c)=
W(tet)
tet=W(m(t))
tet=t
tet=et, using Lemma (2.4) and Corollary (2.6). The fixed
point h(c) = etof g(z) is an attractor. According to Lemma (2.30), |g(h(c))|=|t|<1,
which is true by assumption and the Lemma follows from fixed point iteration theory and
Lemma (2.18).
Lemma 2.32 If |t|>1and c=ϕ(t), then cdoes not exist.
Proof: The fixed point h(c) = etof g(z) is a repeller: According to (2.30), |g(h(c))|=
|t|>1, which is true by assumption and the Lemma follows similarly.
More applications for this section, like Fractals based on repeated exponential iteration
and more identities, may be verified in [210].
2.6 Convergence for qQ
Generalizing the function W = Wcas Whin the space Q, we note that convergence still
happens. First we get a generalization of the Theorem of Shell ([164]).
Lemma 2.33 If qQ, then the sequence nq,nNconverges if t=ϕ1(q)(S3).
Lemma 2.34 If qQand the sequence g(n)(q)converges, the value of the limit is always
given as:
h(q) = Wh(ln(q))
ln(q)
In the space of Qthe function ln is not (in general) commutative, therefore we have
two convergence orbits, however the orbits finally converge to the same point given by
Lemma (2.34) which serves as a generalization of Lemma (2.18).
Examples: If q= 1/21/2i+1/2j1/2kthen |tq|=|−WH(log(q))|.
= 0.70724 <1,
therefore tq(S3), therefore according to Lemma (2.33) the sequence nqconverges. The
15
first 14 values with Maple are given as:
1q.
= 0.50.5i+ 0.5j0.5k
2q.
= 0.34967 0.11655i+ 0.11655j0.11655k
3q.
= 0.75577 0.16732i+ 0.16732j0.16732k
4q.
= 0.51884 0.30319i+ 0.30319j0.30319k
5q.
= 0.49389 0.17222i+ 0.17222j0.17222k
6q.
= 0.63600 0.20888i+ 0.20888j0.20888k
7q.
= 0.53831 0.24422i+ 0.24422j0.24422k
8q.
= 0.54276 0.19809i+ 0.19809j0.19809k
9q.
= 0.58839 0.21696i+ 0.21696j0.21696k
10q.
= 0.55060 0.22511i+ 0.22511j0.22511k
11q.
= 0.55731 0.20924i+ 0.20924j0.20924k
12q.
= 0.57094 0.21767i+ 0.21767j0.21767k
13q.
= 0.55692 0.21898i+ 0.21898j0.21898k
14q.
= 0.56109 0.21373i+ 0.21373j0.21373k
The central Lemma (2.34) gives the limit: q=h(q).
= 0.561820.21640i+0.21640j
0.21640k.
If q= 1 + i+jkthen |tq|=| WH(log(c))|.
= 0.94126 <1, therefore tq(S3),
consequently again based on Lemma (2.33) the sequence nqconverges also. Lemma (2.34)
gives the limit: q=h(q).
= 0.46827 + 0.33791i+ 0.33791j0.33791k.
Details and extensive analysis for the results in the space Qare given in [207].
3 An application of functions HW to infinite exponen-
tials
Divergence in the previous sections doesn’t necessarily mean that |c|=. Divergence
may be such that the limit is , but can also be periodic and even totally chaotic. The
periodic points given by (2.18) all have period 1, so it follows that these are automatically
solutions to the n-th auxiliary equation g(n)(z) = z, but not all solutions are given only
from the main Lemma. Such points exist. To determine such points we need functions
which are stronger than W.
3.1 Definitions
Suppose fi(z) are non-vanishing identically complex functions. We define Fn,m(z) : N2×
CCas:
Definition 3.1
Fn,m(z) = ez, if n= 1,
ecm(n1)Fn1,m (z), if n > 1.
16
Definition 3.2 G(f1, f2, . . . , fk;z) = z·Fk+1,k+1(z)
If k= 0, then G(z) = z·ez. If k= 1 then G(c1;z) = zec1ez. If k= 2 then G(c1, c2;z) =
zec1ec2ez
. When we write about the HW, we can use the terminology G(. . . ;z), meaning
that the corresponding function includes meaningful terms-parameters. The function of
interest here is the inverse of G(. . . ;z).
HW(f1, f2, . . . , fk;y) (9)
In other words Gand HW satisfy the functional relation:
G(. . . ; HW(. . . ;y)) = y(10)
by supposing always that the list of parameters is identical on both sides. These maps
will henceforth be called HW functions. We note that when k= 0, HW(y) = W(y) is the
function W. The existence of the HW is guaranteed in all cases by the Lagrange Inversion
Theorem (see [159, 201-202]).
3.2 The central lemma on HW
A generalization is now immediate, so we are close to a variant of the central Lemma.
Lemma 3.3 If cC\ {0,1}, then the p-th auxiliary equation f(p)(z) = z, admits the
solutions z, f (z), . . . , f(p1)(z), where
z=HW(log(c), . . . , log(c); log(c))
log(c)(pparameters)
Proof: Similarly to the central Lemma, the functions HW solve the p-th equation
g(p)(z) = zas follows:
g(p)(z) = z
cc...cz
=z
z·cc...cz
= 1
z·elog(c)e...elog(c)z
= 1
zlog(c)·elog(c)e...elog(c)z
= log(c)
zlog(c) = HW(log(c), . . . , log(c); log(c))
z=HW(log(c), . . . , log(c); log(c))
log(c)p parameters
We note that if zis a solution of g(p)(z) = z, then if k∈ {1,2, . . . , p 1}we also have
g(p)(g(k)(z)) = g(k)(g(p)(z)) = g(k)(z) and the Lemma follows.
Corollary 3.4 Whenever nc,nNis an attracting cycle, the limits of the psubsequences
n+kc,nN,k∈ {0,1, . . . , p 1}are given as z, g(z), . . . , g(p1)(z), with zbeing given
by the expression of Lemma (3.3).
17
Finally,
Lemma 3.5 If cC\RS T and t= HW(log(c), . . . , log(c); log(c)) (pparameters),
then the sequence nc,nNis an attracting cycle of period p, if tetp2
k=1 f(k)(et)<
log(c)p+1and only if tetp2
k=1 f(k)(et)log(c)p+1.
Proof: The multiplier of the fixed point zof the function g(n)(z) is given in [209], as:
g(n)(z) = (log(c))n·
n
k=1
g(k)(z) (11)
The fixed point zis given by the expression of Lemma (3.3), through the HW function,
so substituting this point in equation (11), the multiplier takes the form:
(log(c))p1·tet
p2
k=1
g(k)(et) (12)
The attractor condition of the function g(p)(z) at the fixed point zis g(n)(z)1,
therefore the Lemma follows from equation (12).
Example: Can we say anything for example about the sequence nc,nN, if c=1+i?
We can use the code of the Appendix to check the measure of the corresponding multipliers
tpfor p-cycles. t1=ϕ1(c) = W(log(c)) .
= 1.13445, therefore it cannot be a cycle
of period 1 (in other words we don’t have convergence). t2.
= 0.80847 + .33448i, and
|t2et2log(c)|.
= 4.67684, therefore it cannot be a cycle of period 2. t3.
= 0.032810.08534i
and |t3et3f(et3) log(c)2|.
=.94227, therefore it has to be necessarily a cycle of period
3. From Corollary (3.4), z= HW(log(c),log(c); log(c))/log(c).
=0.03344 0.01884i,
therefore the three limits of the cycle are given as z, g(z), g (2)(z). This set is calculated
based on the code with accuracy to 5 decimal digits as, 0.03344 0.01884i, 1.02959
0.08808i,1.29112 + 1.19363i. Convergence to these points is spiral like (see [209]). The
HW maps are studied separately and the exact mechanism which extracts the roots is
shown in [205].
4 Topology of the infinite exponential
4.1 Julia and Fatou sets
To study generally (and specifically) the behavior of the sequence of Euler, we study
the corresponding topology which results on the complex plane from repeatedly iterating
the exponential map, for different (but specific each time) base of the exponential, since
Euler’s sequence essentially reduces to repeated iteration of some suitable exponential
map, for a specific base λ= log(c), with cCand initial value z0C.
This last topology is the union of two notable sets of the exponential family, that is
the Julia set J[gc] and the Fatou set F[gc] of the exponential, which are complementary
on the complex plane.
18
To characterize therefore the topology of the sequence of Euler, it suffices to charac-
terize exactly the sets J[gc] and F[gc] of the exponential map. This characterization has
been done however in the references for the family of maps Eλ(z) = λ·ez, with λ= log(c).
Specifically, the Julia set of the exponential map Eλ(z) = λ·ezis given in [76], [75]
and [77] and is the complement of the set of periodic repellers of Eλ(z), that is, J[Eλ] =
zC:E(n)
λ(z)→ ∞. The Fatou set of the exponential map Eλ, is the complement of
the set J[Eλ] or the set of its periodic attractors, that is F[Eλ] = zC:E(n)
λ(z)M.
Proposition 4.1 The Julia and Fatou sets of the topology of the infinite exponential,
coincide with the corresponding sets of the repeated iteration of the exponential map
Eλ(z) = λ·ez, with λ= log(c).
Proof: In the aforementioned references it is proved that the Julia set J[Eλ] of the family
of exponential maps Eλ(z) = λ·ezis a Cantor bouquet for 0 < λ < 1/e, therefore it suffices
to show that in our case the set J[gc] = J[cz] is in general such a bouquet. Note that the
iteration orbit in the references is E=λ·ez, λ ·eλ·ez, . . .. This orbit can be written
alternatively also as E=λ·ez, λ ·(eλ)ez, . . ., therefore setting λ= log(c), the orbit
is equivalent to the orbit Eλ=log(c)=log(c)·ez,log(c)·cez, . . .. For z= 0, we get the
orbit of zero, as E0={log(c),log(c)·c, . . .}. We observe that eE0=(1,)c, cc, ccc, . . .,
which is the simplest orbit of the Euler sequence for the infinite exponential, therefore
the set J[g] will be exactly what the set JEλ=log(c)is, on the complex plane λ= log(c).
Specifically, it is a Cantor bouquet. Since the orbits are coincident modulo λ, the same
will hold also for the Fatou sets and the proof is complete.
4.2 Attractors and repellers
We classify now the topology of specific sub-cases for given λ= log(c) in a neighborhood
Bδ,δ > 0 of the boundary of the Shell-Thron region, by period.
Proposition 4.2 When for λ= log(c)the set J[g]is a Cantor bouquet p-furcation,
then, if there are attractors in the set F[g], they are given from the expression A1=
W(λ)/(λ)if p= 1 or from the expression Ap=gk(z0), z0= HW(...,λ)k
{0,1, . . . , p 1}, if p2.
Proof: The proof is done inductively depending on the value of λ= log(c) on the complex
plane, meaning, depending on the position of c=eλrelative to the region RST .
c(RST Bδ(ϕ(t))). Setting λ= log(c) = log(ϕ(t)) = t/et, by assumption is
0<|t|<1, therefore we get the sub-case 0 <|λ|=|t/et|<1/e. In this case, the
sequence of the infinite exponential converges to A1=h(0, c) = W(λ)λand the
set J[gc(z)] is a Cantor bouquet. The set F[gc(z)] then is the complement of the set
J[gc(z)] on the complex plane and contains exactly one attractor A1, which drives
any point of F[gc(z)] to A1.
c∂RS T Bδ(ϕ(t)). Setting λ= log(c) = log(ϕ(t)) = t/et, by assumption is
|t|= 1, therefore we get the sub-case |λ|=|t/et|. In this case, when p= 1, then
t= 1 and |λ|= 1/e. The set J[gc(z)] then is a plain Cantor bouquet displaying a
19
single fork at the neutral point N1=h(0, ee1) = e. The set F[gc(z)] is simply the
complement of the set J[gc(z)] on the complex plane. When it happens tp= 1 with
p= 2, it also is t=1 and the sequence of the infinite exponential breaks into a
fork of period p= 2 on the complex plane, therefore the set J[gc(z)] consists of two
Cantor bouquets displaying a double fork at the neutral point N2=h(0, ee) = 1/e.
The set F[gc(z)] then, consists of p= 2 sub-regions without attractors. Continuing
inductively, when it is tp= 1, p > 2, the sequence breaks into a fork of period p > 2
at the neutral point N3=h(0, c) = W(λ), therefore the set J[gc(z)] consists
of two Cantor bouquets displaying a p-furcation at the neutral point N3. The set
F[gc(z)] then, consists of p > 2 sub-cases without attractors.
c[(RST )cBδ(ϕ(t))]. Setting λ= log(c) = log(ϕ(t)) = t/et, we get the sub-
case |λ|=|t/et|with |t|>1 and tp= 1. If p2, then the case reduces to
sub-cases p2 of the cases 1 and 2, above. Consequently, if p > 1, the set
J[gc(z)] consists of two Cantor bouquets displaying a p-furcation at the repeller
R1=W(λ)and the set F[gc(z)] consists of psub-areas of the complex plane
(Fatou flower in [78]), with each petal of the flower containing the periodic attractor
Ap=g(k)(z0), z0= HW(. . . , λ),k∈ {0,1, . . . , p 1}, which leads any of the
points in the petals to Ap.
ϕ(t), which delineates the region RST , is conformal at every point of the unit circle
except at the point ϕ(1) = ee1, therefore the three cases above cover fully the boundary
of RST except for a neighborhood Bδ(ϕ(1)), δ > 0, of the point ϕ(1), at which the map
ceases to be conformal. In any neighborhood of this point we can have |λ| ≥ 1/e, and in
such a case, the corresponding set J[gc(z)] suffers periodic explosions, which the references
call Knaster explosions, in which this set suddenly explodes and covers the entire complex
plane in the form of an indecomposable continuum. Otherwise, using δ > 0 we cover fully
the region RST \Bδ(ϕ(1)), therefore we know fully the sets J[gc(z)] and F[gc(z)] and the
proof of the proposition is complete.
4.3 General topological map of the infinite exponential
A general map which describes the local behavior of the sequence αnon the complex plane
for the corresponding sets which will be created using a given base of the exponential
λ= log(c), for cCcan be seen in [93] and [66] and is given here in Figure 3.
The colored regions are areas of period p1. The black pixels are points which have
escaped the size bound that has been set when imaging with the program. Consequently
the accuracy on these regions depends on the aforementioned escape bound set when
iterating the sequence of Euler.
The neutral points of J[g] are precisely the points ϕ(t), with tan n-th root of unity
and ϕ(z) = exp(z·exp(z))), which lie in ∂RST .
The simplest case therefore is given by p= 1 and c=ee1, which is a plain bouquet,
with one conspicuous neutral point at the vertex of the bouquet at h(0, c). The imme-
diately next case is given by p= 2 and c=ee, which is two bouquets interacting at
the neutral point h(0, c). The general case for c=ϕ(t), with tp= 1, are two bouquets
which interact with period pat the neutral point h(0, c). The rest of the cases for |t|>1
give in general two bouquets which interact with period pat h(0, c) in combination with
20
Figure 3: Parameter map of gc(z), cC
the p-periodic attractors Ap, which are vertices of a deformed p-polygon on the complex
plane.
When a point moves from the interior of the region RST to the exterior, during its
passing from ∂RS T , we observe a periodic explosion or a fork of Cantor bouquets, in
which gradually show up pperiodic attractors around the point h(0, c), where pis the
pre-period of the multiplier of c, as in [76] and [78]. In other words, we observe a Knaster
explosion of period p.
The regions of the complex plane where αncauses forks of period pis precisely the
Fatou-like subset F[gc(z)] of the map of Figure 3. This subset consists of infinitely thin
threads which divide the complex plane into regions where αnis periodic. These threads
create Cantor bouquets and their accuracy depends on the accuracy of the calculating
program. Had the accuracy of the calculations been unlimited, the threads of the bouquets
would not show at all, since the thickness of any single thread is actually zero.
Consequently, Lemma (3.3) generalizes the central Lemma (2.18) and classifies nor-
mally the periods of the sequences for any point on the complex plane with sole exception
a neighborhood of ϕ(1). This answers Proposition (1.2) of Euler in its general form, that
is for general x=c̸=ee1on the complex plane.
The characteristic points therefore of the topology of the infinite exponential as it
follows from the above are exactly p+1 points: Complex infinity, towards which move all
Cantor bouquet points and pattractors of the Fatou lake beds (when they exist) as they
are given by Lemma 3.3, towards which move all the points of these lake beds, in spiral
manner.
The torsion angle of the spiral convergence of the periodic attractors of the lake beds
is a function of the neighborhood of the point of the base of the exponential map in
relation to the phase angle of the boundary of the Shell-Thron region. This means that
21
to get spiral convergence with greater or smaller torsion angle, it suffices to change a bit
the angle of the pre-period of the base of the exponential map in a small neighborhood
of the base, relative to the previous phase angle of the base.
4.4 Topology
Proposition (4.2) shows that the topology of the set which results from the repeated
application of the exponential map, when it is not subject to Knaster explosions, is in
general a psymmetry or in other words, a Fatou flower or differently, a deformed p-polygon
on the complex plane or otherwise a fork of Cantor bouquets of period p.
The topology of a Cantor bouquet is given in the references above and is homeomorphic
to the topology of a union of half open intervals [α, +), with the αof the bouquet vertices
in correspondence to the points of a sheared Cantor-like set CS (see section 4.5).
If the bouquet is CBh(k,c)=
k
[αk,+), in the general case, the topology of the
infinite exponential, will be J[gc(z)] F[gc(z)] = CBh(k,c)× ⋄p. In other words, a fork
of Cantor bouquets, of period p.
Examples are given in Figures 5 and 8, 9, 10, 11, 12, 13 of the Appendix.
4.5 Fractal nature in the infinite exponential
The Julia sets here, that is the Cantor bouquets of the infinite exponential display an
intense fractal character. Specifically, for given screen resolution j, which is a function of
the bail-out value in the example of the code in the Appendix, Devaney represents them
using “fingers” δj
n,nN.
We note a periodic break-up of each finger δjinto a subsequence of fingers δj+1
n,nN,
for each apparent finger δj, which repeats ad infinitum for each new finger. Consequently
the following relation gives a partial cover of the threads of a bouquet at resolution level
j, as δj=
n=0 δj+1
n. An example is given in Figure 4.
Figure 4: Devaney “finger” δjof Cantor bouquet
In an orthocentrically oriented finger δj, like that of figure 4 (oriented in this case with
infinity down), the vertical width of the bouquet is bounded (see again [76], [75] and [77])
and the break up of half the finger can be set to a homeomorphic correspondence with
a suitable Cantor set, on the closed vertical interval [a, b], with athe vertical projection
22
of the vertex of δjon the real axis and bhalf the horizontal dimension of the finger.
The central lines in Figure 4 show approximately the the homeomorphic correspondence
implied for finger δj.
The dimension on either axis is additive, so if anand bn,nNare the traces of the
bases of the (sub-)fingers δj+1 on the interval [a, b], we write the recursive expression that
describes the break up of finger δjinto fingers δj+1 as,
dimH,x(δj)
j=0
dimH(δj+1)x=
n=0 |bnan|(13)
The total thickness of the finger is finite (2 |ba|) in the horizontal dimension, there-
fore the sequence dimH,x(δn)n
j=0 |bnan|is geometric with ratio,
r=
dimH,x(δn+1 )
dimH,x(δn)=δn+1 x
|δn|x
(14)
Note that ris independent of n(being the fixed ratio of a geometric), therefore
equation (14) must be satisfied also for n= 1, which reveals that in the given resolution
of the finger δjwe have in our screen it suffices to measure the difference in the diameters
of any two successive relative to resolution level fingers in the image of the bouquet.
If we had then at our disposal a “straight” bouquet against at least one dimension x
or y, then it would suffices to measure the difference in diameter between any two such
fingers in screen pixels and we would immediately get the ratio of the geometric that is
the break up ratio of the Cantor bouquet by at least one direction.
There exists one and only one such bouquet. The bouquet produced by the exponential
map with base λ=e1, which sees at point exp(1) (preperiod p= 1, center), which
includes the interval [e, +). For any other base apart from this lambda the behavior
of the exponential for a base in the right or left neighborhood of this point (λ) in the
direction θis unpredictable with Knaster explosions which deform the bouquets and force
them to suddenly cover the entire complex plane. Such a measuring of difference between
fingers therefore is not possible in images of bouquets for base |λ|>1/e.
Yet we also have the images from all the other values of the base we calculated. No
bouquet however allows accurate measuring by specific dimension, because the bouquet
threads may suddenly turn unexpectedly towards a direction other than the expected one
we measure in, therefore any such measuring in this direction will necessarily produce
errors.
This way, the only bouquet available is that of λ= 1/e, since it’s the only one among
the one’s we’ve described, for which we know that exactly one of its threads is a straight
line. An image of this bouquet is given in Figure 5. The image of this bouquet however
extends all the way to positive infinity (case of angle θ= 0), therefore it distorts the ratio
rof the geometric at least by direction θ.
The actual ratio of the geometric therefore can be measured in direction xewith
the help of the map µ(z) = 1/z, finding the inverse image relative to infinity as a fractal
set using some Maple code. It is the image which shows in Figure 6.
µis meromorphic in any neighborhood of infinity |z| ≥ R > 0 and the exponential
with base λ=e1is also meromorphic on the interval [e, +), therefore the image of
Figure 6 is conformal on [µ(R), µ(e)].
23
Figure 5: Cantor bouquet at c=e(of base λ=e1)
Figure 6: Cantor bouquet at c=eunder the map µ(z) = 1/z
Therefore on this interval the dimension is given by the topology of the measure of the
absolute value of a geometric with successive finger intervals an+1 =bn, for each nN,
as
24
|µ(e)µ(R)|x=
n=0
dimH,x(δn)
=
n=0 |δn|x
=
n=0 |bnan|
=
n=0 |b0a0| · rn=|b0a0| · 1
1r
(15)
Considering now a sequence Rn→ ∞, the measure of the geometric can be measured
experimentally from Figure 6, either as,
rx= lim
Rn→∞ 1δ0x
|µ(e)µ(Rn)|<1 (16)
or as,
rx=dimH,x(δ2)
dimH,x(δ1)=δ2x
|δ1|x
<1 (17)
Using equation (17) in2Figure 6, we find,
rx
490 337
337 150= 0.8181 (18)
In the horizontal direction we therefore have dimH,x(δ)rx. In the vertical direction
(y) the dimension of the bouquet is at least that of the continuum ([e, +)) therefore,
1dimH,y(δ)
0dimH,x(δ)rx
1dimH(δ)1 + rx= 1.818
We finally get 1 dimH(CBe)1 + rx= 1.8181, which means3that the dimension
of the Cantor bouquet at e(λ=e1) is fractional with geometric ratio of decomposition
rby at least one arbitrary direction.
Against the vertical direction then, the intersection of the bouquet λ= 1/e is home-
omorphic to a generalized4Cantor set, of dimension r, which is a fractal. The dimension
of such sets is analyzed partly in [60] and is given as a function of a general parameter
0< γ < 1, as
2For the measuring a figure of greater resolution was used.
3The java program on page “http://www.stevec.org/fracdim/” on the net gives r1.85 using the
box counting topology.
4Generalized but symmetric Cantor set, as the bouquet is symmetric relative to the axis [e, +).
25
r=ln(2)
ln(1γ
2)(19)
In our case we have then r0.8181, therefore if we solve equation (17) for γ, we get
γ1/7, which means that the corresponding sheared set is homeomorphic to a Cantor
set which has a geometric break down ratio 1/7. The common known Cantor set has a
break down ratio equal to 1/3. We can therefore say that the bouquet at efor λ= 1/e
is a fractal. It is dense by vertical section and the set CBe{+∞} is connected. If we
remove however the point {+∞}, then the set is totally disconnected.
4.6 The primitive of a Cantor bouquet
The primitive of a Cantor bouquet therefore may be revealed if we examine the image
of the n-th roots of the neighborhood Bδ(ϕ(1)), under the map gc(z) and is presented
here in Figure 7 for n= 100, of the neighborhood δ= 0.01, of ϕ(1) = ee1. The Figure
validates the aforementioned references which characterize it as homeomorphic to a C
brush.
Figure 7: Primitive of Cantor bouquet at c=e1/e
We conclude immediately that in as far as the bouquet for λ= 1/e is concerned, all
the threads of this bouquet are Ccurves on the complex plane.
The form of the primitive in combination to the measure of the fractal dimension,
reveals fully the structure of the bouquet on the microscope. What is visible elementarily
is in general a smooth (infinitely differentiable) curve-thread, which periodically explodes
into a continuum of other infinitely smooth curves-threads.
The noteworthy characteristic with the threads of any bouquet then, is that only the
coordinates of the thread [e, +) of the bouquet with λ= 1/e are known. Any other
thread is (computationally) unknown, meaning that no point (of any other thread) can
be determined exactly5, except the points towards which the bouquets “point” when we
5Except as a function of the resolution level of the screen jor equivalently of the bail-out value in the
code of the Appendix
26
have a bouquet fork of period p1. In contrast with the previous characteristic, the
attractors and the repellers of the Fatou lake beds can always be expressed analytically.
4.7 Pinball with the infinite exponential
We also observe that the whole topology of the infinite exponential is additionally periodic
modulo 0 < θ 2πon the complex plane, as all figures display bouquets which protrude
against the center under a specific angle θ. This angle is characteristic of the angle of the
pre-period of the base λof the exponential.
The previous periodicity allows us to concentrate on the topology of the Euler sequence
on the complex plane modulo one turn θwhich is finally equivalent to the dynamics of the
orbits of a ball in a pinball game on the strip {z:|ℜ(z)| ≤ bθ} × {−∞}, equipped with
the topology of the Fatou lake beds and highlands or Julia crevices with metric the speed
of convergence/attraction (also see [87]) or repulsion to infinity and with the attractor at
infinity being equivalent to gravity on a pinball table of angle θ.
Whatever is close to a Cantor bouquet sooner or later goes to complex infinity (side
direction of gravity under plane angle θrelative to horizontal plane), while whatever
is inside a lake bed finally falls inside a small hole like in a regular pinball machine of
inclination θ, with plocal holes or dingy surface ornaments.
With the exception of a neighborhood of the point λ= log(c) = e1, for rational
multiples of the pre-period angle 2π/p of the base λof the exponential map gc(z) = cz,
the topology of the infinite exponential is (at least) a Cantor bouquet at a p-furcation in
a Fatou lake bed with center α, with αand pNdetermined analytically.
5 The generalized infinite exponential
5.1 Definitions
The general infinite exponential is analyzed in a limited way in [210] ([18],[178], [179])
and in [122]. It is defined recursively:
Definition 5.1 If Zk={z1, z2, . . . , zk},kN, with zkC\ {xR:x0}and n
|Zk|=k,
en(Zk) = zk, if n= 1,
zen1(Zk)
kn+1 , if n > 1.
Whenever we need to study convergence of the previous exponential, we simply ex-
amine the values of the sequence bn=en(Zn). Unwinding recursively bn,nN, using
Definition (5.1), produces the terms: z1, zz2
1, zzz3
2
1, . . . , zzz···zn
3
2
1.
There is however a dual definition which reverses the indexes, as
Definition 5.2 Given a list Zk={z1, z2, . . . , zk},kN, with zkC\ {xR:x0}
and n≤ |Zk|=k,
e
n(Zk) = z1, if n= 1,
ze
n1(Zk)
n, if n > 1.
27
Analyzing recursively the sequence b
n=e
n(Zn), nN, using Definition (5.2), pro-
duces the sequence z1, zz1
2, zzz1
2
3, . . . , zzz···z1
n2
n1
n.
Whenever the general infinite exponential converges, we write,
Definition 5.3 Whenever the following limit exists finitely,
e(Z) = lim
n→∞ en(Zn) = lim
n→∞ bn=b
e
(Z) = lim
n→∞ e
n(Zn) = lim
n→∞ b
n=b
The references ([18]) show the following (in relation to Definitions (5.1) and (5.2)):
Theorem 5.4 (Barrow) The sequence bnconverges, if and only if:
• ∃n0:nn0:bn[(1/e)e, e(1/e)].
znconverges.
Theorem (5.4) can be now generalized, since we now know the full extent of the RST
region:
Theorem 5.5 (Barrow) Sequence bnconverges, if and only if:
• ∃n0:nn0:bnRST .
znconverges.
We are interested in finding interesting expressions for the limits band b
.
5.2 Ascending indexes
For the first case (Definition (5.1)), we set Gn(z) = e
n+1(Ln {z}), in which case we get
the iteration:
Gn(z) = y
log(Gn(z)) = log(y)
Gn1(z) = log(y)
log(zn)
log(Gn1(z)) = log log(y)
log(zn)
Gn2(z) = log(Gn1(z))
log(zn1)
·· ·
Gnk+1(z) = log(Gnk(z))
log(znk)
Gm+1(z) = log(Gm(z))
log(zm)
(20)
28
Suppose z0= lim
n→∞ zn. We select ksuch that m=nkn0. We can take limits for
m→ ∞ in the last of (20) and solve for W:
G(z) = log(G(z))
log(z0)
G(z) = W(log(z0))
log(z0)=h(z0)
(21)
Equations (21) show,
Theorem 5.6 If z0= lim
n→∞ zn, then for ϵ > 0, there exists n0, such that for nn0,
|b
nh(z0)|< ϵ.
Proof: For any ϵ > 0, we can find n0sufficiently big that guarantees |znz0|< ϵ and
then the bases of the tower until zn0can be expressed as z0+ϵ. The elements further
than zn0can be gradually sent to infinity, by choosing big n0and then the limit of the
tower will approach the number h(z0+ϵ). The Theorem follows by forcing ϵ0.
5.3 Descending indexes
For the dual case (Definition (5.2)), if Zk={z1, z2, . . . , zk}, transforming as wm=
znm+1 and as Z
k={w1, w2, . . . , wk}to change the order of the indexes, we set Gn(z) =
e
n(Z
n∪ {z}), in which case we get the iteration:
Gn(z) = y
log(Gn(z)) = log(y)
Gn1(z) = log(y)
log(wn)
log(Gn1(z)) = log log(y)
log(wn)
Gn2(z) = log(Gn1(z))
log(wn1)
·· ·
Gnk+1(z) = log(Gnk(z))
log(wnk)
Gm+1(z) = log(Gm(z))
log(wm)
(22)
Suppose z0= lim
n→∞ zn. We choose ksuch that m=nkn0. Similarly, we can get
the limit for m→ ∞ in the last of (22) and to solve for W:
G(z) = log(G(z))
log(z0)
G(z) = W(log(z0))
log(z0)=h(z0)
(23)
29
Equations (23) show,
Theorem 5.7 If z0= lim
n→∞ zn, for each ϵ > 0, there exists n0, such that for each nn0,
|bnen0+1(Zn0∪ {h(z0)})|< ϵ.
Proof: Similarly, given ϵ > 0, we can find n0big enough that guarantees |znz0|< ϵ,
and then all the terms higher than zn0can be approximated as h(z0+ϵ). The tower below
zn0, leaves the elements untouched, therefore the approximation will be zz···
zh(z0)+ϵ
n0
2
1. The
Theorem follows by forcing ϵ0.
Theorems (5.6) and (5.7) in combination with Proposition (4.2) answer Proposition
(1.2) of Euler in its general form. Specifically,
Corollary 5.8 Suppose the sequence γn,nNwith γncC. The behavior of the
sequence αnwith αn+1 =γαn
n,nN, is described by the sets J[gc+ϵ]and F[gc+ϵ], with
ϵ > 0from Theorem (5.6).
30
6 Appendix: programming/results
Executable Maple code is attached in a separate .pdf document, to which the author refers
for the validation of the Corollaries and the creation of the Figures of the dissertation. In
the figures, the first column contains an attractor A1=h(0, c), the second column contains
the neutral point N1=h(0, c) and the third column contains the repeller R1=h(0, c)
and the periodic attractors Apof Proposition (4.2), distributed around the repeller R1.
Figure 8: Preperiod p= 1, with λ= log(c) = t/et,|t|<1, t= 1 and |λ|>1/e
Figure 9: Preperiod p= 2, with λ= log(c) = t/et,|t|<1, t=eπ i =1 and |t|>1
Figure 10: Preperiod p= 3, with λ= log(c) = t/et,|t|<1, t=e2π/3iand |t|>1
31
Figure 11: Preperiod p= 4, with λ= log(c) = t/et,|t|<1, t=eπ /2iand |t|>1
Figure 12: Preperiod p= 5, with λ= log(c) = t/et,|t|<1, t=e2π/5iand |t|>1
Figure 13: Preperiod p=π, with λ= log(c) = t/et,|t|<1, t=e2iand |t|>1
32
7 Acknowledgements
The author wishes to thank all the factors who contributed in the exposition of the work.
Specifically, his advisor and professor Yiannis Papadoperakis for the almost unbeliev-
able patience he showed during the routing of the dissertation, professor Paris Pamfilos
and professor Michalis Lambrou for their recommendation letters and all the factors that
solved many subproblems, like the researcher Alexandr Dubinov who solved Kepler’s
equation, the professors of the University of Crete, the advisors and mathematicians of
the internet who answered many of the questions in international forums and usenet,
like professor Robert B. Israel et al., the mathematics programmers of the Maple forum
et al., Leroy Quet who solved the problem of the series development of the exponential
(Lemma (8.1) in [210]), Daniel Geisler of the page tetration.org, Henryk Trappman, An-
drew Robbins, Gottfried Helms et al., of the tetration forum, Eric Weisstein of the page
mathworld.wolfram.com the referees of the author’s articles, professor Dave L. Renfro who
collected, archived and sent most of the references and the creators of the used fractal
programs which validated the results of the work, like Jan Hubicka and Thomas Marsh.
More acknowledgements can be found in the author’s references. The author dedicates
the dissertation to his parents and to his first mathematics professor Kostas Sourlas.
33
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