In the problem of finding fixedThe following is about getting help for a proof on existence and indexability of periodic points of the complex exponential-function I introduced the notation for the function of a vectorial argument $K$
$$ T_n=\text{Find}(K) \qquad \text{with } K=[k_1,k_2,...,k_n] , k_i \in \mathbb Z
$$
Here the function $\text{Find}()$ applies the nested logarithm, here with given branchindexes $k_i$ to some initial value and iterates a handful of times until it stops when an accuracy-criterion is reached and gives back a vectorbase $T_n$ of$e:=\exp(1)$.
Update The question is a complete rewriting of the previous formulation of my question which I hope is much better focused and straightforward.
Let us define $n$-periodic points of one specific period in the sense$f(z):=\exp(z)$ for $T_n[1]=\exp(T_n[n])$$z \in \mathbb C$. Iteration may be denoted by $f°^1(z)=f(z)$ and $T_n[k-1]=\exp(T_n[k])$$f°^{h+1}(z)=f°^{h}(f°^1(z))$.
An unoptimized Pari/GP-codeFixpoints: it is known (final Newton-iteration for arbitrary precision omittedfor instance W. Bergweiler1, pg 16),
c = I*2*Pi;
{Find(K)=my(T,n=#K,maxit=20);
T=vector(n);T[1]=1+I; \\ some initial value, but remember the complex conjugacy
for(it=1,maxit,
for(j=1,n,
T[(j % n)+1]=log(T[j])+K[j]*c
);
);
return(T); } \\ returns a vector of length n containing the members of one period
- that $f$ has infinitely many fixpoints (=$1$-periodic points) $p_1$,
- that all of them are non-real, and
- that all of them are repelling.
Using the exponential function itself for finding periodic points is difficult becauseThey may be indexed by the fixed and periodic points are all repelling and to usebranchindex $k \in \mathbb Z$ used in the NewtonLambert-algorithm for iteration towards the searched periodic point one needs to be very near to it, with distance of less than, say,$W(k;z)$-function like $1e-5$$p_{1:k}$.
But using the logarithm instead we have attraction in the full complex planePeriodic points: it is further known that for (except atall $z \in \{0,1,e,e^e,...\}$) to one of$n$ the sets of $n$-periodic points are as well infinite1(conjugatedpg.16) fixpoints $t_1 \approx 0.3181 \pm 1.3372 î $. $10$ or $20$ iterations of the logarithm give enough approximation to successfully follow up with Newton-iteration to get machine-possible precisionLet us denote such a set from now $\mathbb P_n$. So, in generalization of the above rough Pari/GP-procedure we would simply append a polishingindexing of $1$-periodic points, one might say, that the resultset $\mathbb P$ of all $\mathbb P_n , \text{with } n=1 \ldots \infty $ can be indexed by the Newton-approximation method$\mathbb Z^\infty$. My Pari/GP function would give with
But I think this $T_1=\text{Find}([0])$is not precise enough; I assume, and $t_1=T_1[1] $ numericallywould like to, say prove, five digits correct to the above given value.
- (Conj. 1) that actually $\mathbb P_n$ can exactly be indexed by $\mathbb Z^n$ $\qquad \Leftarrow \mathbb {\text{problem to be proved}}$
If we use the branches in the application ofFor my approach arguing for (Conj. 1) I refer to the complex logarithm like $\mathbb T_1: \{T_1=\text{Find}([k]) \qquad k \in \mathbb Z\}$ we find furtherproperty, that a fixed (=$1$-periodic) or periodic points $t_1$, which is repelling for eachiteration over function $k$.
There are infinitely many of them$f()$, as provedis attracting for instance by Hellmuth Kneser$\,^{[1]}$ in his treatize oniteration over its inverse, which means is attracting for the fractional iteration ofover the exponential function $\log()$-function. MoreoverFrom Bergweiler, Kneser proved alsopg. 17, I take, that the set of fixed ($1$-periodic)all periodic points $\exp(t_1)=t_1$ agrees with the set of branched logarithms, meaning the finding of fixed-points byare repelling on iteration of the branched logarithmon $\mathbb T_1$ is exhaustive. This is done by considering the real$f()$ and imaginary components of the pointthus are $t_1 = a+ bî = \exp(a+bî) $ and stating equationsall attracting for $a$ and $b$ whose set of solutions is equivalent to the set of branch-indexes of the $\log()$-functionits inverse.
MyFor convenience of further notation let us define $T_n=\text{Find}(K)$ function generalizes this to$\log()$ as $n$-periodic points. I have given some examples in my$g(x):=\log(x)$ and as well the iteration MSE-Q&A for$g°^1(z)=g(z)$ and $n = 2,3,5,13,31$ periods$g°^{h+1}(z)=g°^{h}(g°^1(z))$.
Update: I should perhaps To make it clearer, that one $T_n$ contains the $n$ elements$g()$ a true inverse of one specific $n$-period$f()$, such that for instance
we'll need the branch index(es) explicite, so let us simply extend the notation
$$\small {\qquad \vdots \\ ...,\text{Find}([-1,-2]), \text{Find}([-1,-1]), \text{Find}([-1,0]), \text{Find}([-1,1]), \text{Find}([-1,2]), ... \\ ...,\text{Find}([0,-2]), \text{Find}([0,-1]), \text{Find}([0,0]), \text{Find}([0,1]), \text{Find}([0,2]), ... \\ ..., \text{Find}([1,-2]), \text{Find}([1,-1]), \text{Find}([1,0]), \text{Find}([1,1]), \text{Find}([1,2]),...\\ \qquad \vdots \\}$$
$$g(x,k):=\log(x) + k \cdot C \qquad \text{where } C=2 \pi î$$
is an expliciteThis allows to make precisely for some fixed (partial) enumeration of$z$
$$ g(f(z),0)=z $$
but for the whole setreversion of some $2$-periodic$z'=z + k\cdot C$ we need
$$ g(f(z'),k)=g(f(z+k\cdot C),k) =g(f(z),k)=z+k \cdot C=z' $$
For adressing periodic points of period-length $$\mathbb T_2 : \{T_2=\text{Find}([k_1,k_2]) \qquad k_1,k_2 \in \mathbb Z\} $$
and in general$n$ we expand the notation further
$$\mathbb T_n : \{T_n=\text{Find}([k_1,k_2,...,k_n]) \qquad k_1,k_2,...,k_n \in \mathbb Z\} $$$$ \begin{array}{} g(z,[k_1])&:= g(z,k_1) \\
g(z,[k_1,k_2])&:= g(g(z,k_1),k_2) \\
g(z,[k_1,k_2,...,k_n])&:= g(...g(g(z,k_1),k_2)...,k_n) \\
&\small \text{where all $k_j \in \mathbb Z$}\\
\end{array}$$
Finally I use $K_n:=[k_1,k_2,...,k_n]$ for the vector of branch-indexes.
With this I conjecture now the following:
- iterations of each expression $z_{i+1}=g(z_i,K_n)$ are attracting.
- we can approximate any periodic point $p_{n,K}$ by simple fixed-point iteration of the previous with some suitable initial value $z_0 \ne 0$ according to $$p_{n,K} = \lim_{i \to \infty} z_{i+1}=g(z_i,K_n)$$ (of course we can increase speed of approximation when Newton-iteration on $g()$ follows).
- the iteration for a given $K_n$ is attracting over the whole complex plane except for the initial values $z_0 \in \{0,1,e,e^e,...\}$.
Non uniqueness occurs only for $K=[0]$ (and its non-primitive repetitions $K=[0,0]$,... $K=[0,0,...,0]$) in that the initial value $z_0$ is relevant for to converge towards the $1$-periodic point either in the upper or in the lower half plane.
- Main conjecture to be proved: All $n$-periodic points with the exception of the conjugated primary fixed points $p_{1:[0]}$ and $\overline {p_{1:[0]}}$ (which have the same branch index-vector $K=[0]$) are in bijection to the indexes $K_n$ and can be approximated by simple fixed-point iteration over $g(z,K_n)$ (if desired followed by Newton-iteration on $g(z,K_n)$ to speed up convergence).
end updateRemark: I have seen, that with exponential bases different from $e:=exp(1)$ spuriously non-uniquenesses and non-existences of $n$-periodic points occur, which I cannot yet nail down except by giving a couple of heuristic examples. However, large surveys on the exponential with base $e$ -as discussed here- seem to have only that one exception as mentioned in (3.).
However I need a proof forAn illustration of periodic points of periods $n=1..5$ . Those were found by screening the claim, thatsquare $-4-4î...4+4î$ on the sets $\mathbb T_n$ are as well exhaustive ascomplex plane in the case ofsteps by $\mathbb T_1$,$1/40$ with the set ofnewton-iteration applied. The list has then been checked whether they all agree with the $1$$K_n$-periodicindexing scheme; all found periodic points have a valid $K$-index.
Of course I tried to generalize the ansatz of Kneser
A handful of used literature: I've found some resources on fixed points and their properties for the exponential function base $2$-periodic case$e$, but it doesn't seem to reach anywhere. Moreover, forless about $n \gt 2$ any attempt in the spirit of the Kneser$n$-proof surely must die because of complexity ofperiodic points. The most fruitful so far was the connected $\sin()$habilitation of Walter Bergweiler,$\cos()$ 1991. If there are more comprehensive texts (optimally online available),$\exp()$-components please leave a comment.
On the other hand1Bergweiler, Walter, it seems vaguely to me that there might be an argumentPeriodische Punkte bei der Iteration ganzer Funktionen, that the cases of higher $n$ should in some wayAachen: Rheinisch-Westfälische Techn. Hochsch., Math.-Naturwiss. Fak., Habil.-Schr. 51 S. inherit their exhaustiveness from the basic case(1991). $\mathbb T_1$, perhaps shown by inductionZBL0728.30021.
But I'm unable to see a path do do thatPg.16:
So my question: How could I proceed to prove (or disprove!) the exhaustiveness of my function $T_n=\text{Find}(K)$ for the finding of $n$-periodic points in the complex $\exp()$ - function?
- "Dieses Ergebnis wurde im Jahre 1948 duch Rosenbloom verallgemeinert, der zeigte, daß für jedes $n \gt 2$ unendlich viele periodische Punkte der Periode $n$ existieren"
- "Baker im Jahre 1960 (...) bewies (...), daß höchstens eine (von $f$ abhängige) natürliche Zahl $n$ existiert mit der Eigenschaft, daß $f$ nur endlich viele periodische Punkte der primitive Periode $n$ hat."
Pg. 17:
- "Satz 2: Es sei $f$ eine ganze transendente Funktion und es sei $n \ge 2$. Dann hat $f$ unendlich viele abstoßende periodische Punkte der primitiven Periode $n$."
- "Wir bemerken noch, daß ganze Funktionen keine anziehenden periodischen Punkte zu haben brauchen. Ein Beispiel, (...) ist durch $f(z)=e^z$ gegeben."
$\,^{[1]}$ Hellmuth KneserAdditional readings:
Hellmuth Kneser, Real analytic solutions of the equation $φ(φ(x))=e^x$ and related functional equations. (Reelle analytische Lösungen der Gleichung $φ(φ(x))=e^x$ und verwandter Funktionalgleichungen.)[Real analytic solutions of the equation $φ(φ(x))=e^x$ and related functional equations. (Reelle analytische Lösungen der Gleichung $φ(φ(x))=e^x$ und verwandter Funktionalgleichungen.)], J. Reine Angew. Math. 187, 56-67 (1949)(German).Zbl 0035Zbl0035.04801.3
Additional readings:
Shen, Zhaiming; Rempe-Gillen, Lasse, The exponential map is chaotic: an invitation to transcendental dynamics, Am. Math. Mon. 122, No. 10, 919-940 (2015). ZBL1361.37002.4
Here general aspects of the set of $n$-periodic points are presented in existence-theorems. Even the concept of infinite non-periodic, but not diverging-to-infinity, orbits -as part of the general chaotic behaviour- is covered by the list of theorems.Here general aspects of the set of $n$-periodic points are presented in existence-theorems. Even the concept of infinite non-periodic, but not diverging-to-infinity, orbits -as part of the general chaotic behaviour- is covered by the list of theorems.(G.H.)
An introductory article which deals with the question of $T_1$$\mathbb P_1$ (fixed-) points on the branches of the $\log()$-function by Stanislav Sykora (2016) at his web-space.
