# Is my ansatz for finding $n$-periodic-points of the exponential-function exhaustive?

The following is about getting help for a proof on existence and indexability of periodic points of the exponential-function, here with base $$e:=\exp(1)$$.

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 $$f(z):=\exp(z)$$ for $$z \in \mathbb C$$. Iteration may be denoted by $$f°^1(z)=f(z)$$ and $$f°^{h+1}(z)=f°^{h}(f°^1(z))$$.

Fixpoints: it is known (for instance W. Bergweiler1, pg 16),

• 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.

They may be indexed by the branchindex $$k \in \mathbb Z$$ used in the Lambert-$$W(k;z)$$-function like $$p_{1:k}$$.

Periodic points: it is further known that for all $$n$$ the sets of $$n$$-periodic points are as well infinite1(pg.16). Let us denote such a set from now $$\mathbb P_n$$. So, in generalization of the indexing of $$1$$-periodic points, one might say, that the set $$\mathbb P$$ of all $$\mathbb P_n , \text{with } n=1 \ldots \infty$$ can be indexed by $$\mathbb Z^\infty$$.
But I think this is not precise enough; I assume, and would like to prove,

• (Conj. 1) that actually $$\mathbb P_n$$ can exactly be indexed by $$\mathbb Z^n$$ $$\qquad \Leftarrow \mathbb {\text{problem to be proved}}$$

For my approach arguing for (Conj. 1) I refer to the property, that a fixed- or periodic points, which is repelling for iteration over function $$f()$$, is attracting for iteration over its inverse, which means is attracting for the iteration over the $$\log()$$-function. From Bergweiler, pg. 17, I take, that all periodic points are repelling on iteration on $$f()$$ and thus are all attracting for its inverse.

For convenience of further notation let us define $$\log()$$ as $$g(x):=\log(x)$$ and as well the iteration $$g°^1(z)=g(z)$$ and $$g°^{h+1}(z)=g°^{h}(g°^1(z))$$.

To make $$g()$$ a true inverse of $$f()$$, we'll need the branch index(es) explicite, so let us simply extend the notation $$g(x,k):=\log(x) + k \cdot C \qquad \text{where } C=2 \pi î$$ This allows to make precisely for some fixed $$z$$ $$g(f(z),0)=z$$ but for the reversion of some $$z'=z + k\cdot C$$ 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 $$n$$ we expand the notation further $$\begin{array}{} g(z,[k_1])&:= g(z,k_1) \\ g(z,[k_1,k_2])&:= g(g(z,k_1),k_2) \\ g(z,[k_1,k_2,...,k_n])&:= g(...g(g(z,k_1),k_2)...,k_n) \\ &\small \text{where all k_j \in \mathbb Z}\\ \end{array}$$

Finally I use $$K_n:=[k_1,k_2,...,k_n]$$ for the vector of branch-indexes. With this I conjecture now the following:

1. iterations of each expression $$z_{i+1}=g(z_i,K_n)$$ are attracting.
2. 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).
3. 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.
4. 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).

Remark: 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.).

An illustration of periodic points of periods $$n=1..5$$ . Those were found by screening the square $$-4-4î...4+4î$$ on the complex plane in steps by $$1/40$$ with the newton-iteration applied. The list has then been checked whether they all agree with the $$K_n$$-indexing scheme; all found periodic points have a valid $$K$$-index.

A long & wide discussion (using other bases than $$e$$, and using another ansatz for partial solutions) can be found at MSE

A handful of used literature: I've found some resources on fixed points and their properties for the exponential function base $$e$$, but less about $$n$$-periodic points. The most fruitful so far was the habilitation of Walter Bergweiler, 1991. If there are more comprehensive texts (optimally online available), please leave a comment.

1Bergweiler, Walter, Periodische Punkte bei der Iteration ganzer Funktionen, Aachen: Rheinisch-Westfälische Techn. Hochsch., Math.-Naturwiss. Fak., Habil.-Schr. 51 S. (1991). ZBL0728.30021.

Pg.16:

• "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."

Additional 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.)], J. Reine Angew. Math. 187, 56-67 (1949)(German).Zbl0035.04801.4

Shen, Zhaiming; Rempe-Gillen, Lasse, The exponential map is chaotic: an invitation to transcendental dynamics, Am. Math. Mon. 122, No. 10, 919-940 (2015). ZBL1361.37002.5
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 $$\mathbb P_1$$ (fixed-) points on the branches of the $$\log()$$-function by Stanislav Sykora (2016) at his web-space.