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In the problem of finding fixed and 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 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 vector $T_n$ of $n$-periodic points of one specific period in the sense $T_n[1]=\exp(T_n[n])$ and $T_n[k-1]=\exp(T_n[k])$.

An unoptimized Pari/GP-code (final Newton-iteration for arbitrary precision omitted)

   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

Using the exponential function itself for finding periodic points is difficult because the fixed and periodic points are all repelling and to use the Newton-algorithm for iteration towards the searched periodic point one needs to be very near to it, with distance of less than, say, $1e-5$.

But using the logarithm instead we have attraction in the full complex plane (except at $z \in \{0,1,e,e^e,...\}$) to one of the (conjugated) 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 precision. So in the above rough Pari/GP-procedure we would simply append a polishing of the result by the Newton-approximation method. My Pari/GP function would give with this $T_1=\text{Find}([0])$ and $t_1=T_1[1] $ numerically to, say, five digits correct to the above given value.

If we use the branches in the application of the complex logarithm like $\mathbb T_1: \{T_1=\text{Find}([k]) \qquad k \in \mathbb Z\}$ we find further fixed (=$1$-periodic) points $t_1$ for each $k$.
There are infinitely many of them, as proved for instance by Hellmuth Kneser$\,^{[1]}$ in his treatize on the fractional iteration of the exponential function. Moreover, Kneser proved also, that the set of fixed ($1$-periodic) points $\exp(t_1)=t_1$ agrees with the set of branched logarithms, meaning the finding of fixed-points by iteration of the branched logarithm $\mathbb T_1$ is exhaustive. This is done by considering the real and imaginary components of the point $t_1 = a+ bî = \exp(a+bî) $ and stating equations for $a$ and $b$ whose set of solutions is equivalent to the set of branch-indexes of the $\log()$-function.


My $T_n=\text{Find}(K)$ function generalizes this to $n$-periodic points. I have given some examples in my MSE-Q&A for $n = 2,3,5,13,31$ periods.

Update: I should perhaps make it clearer, that one $T_n$ contains the $n$ elements of one specific $n$-period, such that for instance
$$\small {\qquad \vdots \\ ...,\text{Find}([-1,-2]), \text{Find}([-1,-1]), \text{Find}([-1,0]), \text{Find}([-1,1]), \text{Find}([-1,2]), ... \\ ...,\text{Find}([0,-2]), \text{Find}([0,-1]), \text{Find}([0,0]), \text{Find}([0,1]), \text{Find}([0,2]), ... \\ ..., \text{Find}([1,-2]), \text{Find}([1,-1]), \text{Find}([1,0]), \text{Find}([1,1]), \text{Find}([1,2]),...\\ \qquad \vdots \\}$$
is an explicite (partial) enumeration of the whole set of $2$-periodic points $$\mathbb T_2 : \{T_2=\text{Find}([k_1,k_2]) \qquad k_1,k_2 \in \mathbb Z\} $$ and in general $$\mathbb T_n : \{T_n=\text{Find}([k_1,k_2,...,k_n]) \qquad k_1,k_2,...,k_n \in \mathbb Z\} $$. end update

However I need a proof for the claim, that the sets $\mathbb T_n$ are as well exhaustive as in the case of $\mathbb T_1$, the set of $1$-periodic points.
Of course I tried to generalize the ansatz of Kneser for the $2$-periodic case, but it doesn't seem to reach anywhere. Moreover, for $n \gt 2$ any attempt in the spirit of the Kneser-proof surely must die because of complexity of the connected $\sin()$,$\cos()$,$\exp()$-components.

On the other hand, it seems vaguely to me that there might be an argument, that the cases of higher $n$ should in some way inherit their exhaustiveness from the basic case $\mathbb T_1$, perhaps shown by induction.

But I'm unable to see a path do do that.

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?


$\,^{[1]}$ 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)Zbl 0035.04801

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. 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.

An introductory article which deals with the question of $T_1$ (fixed-) points on the branches of the $\log()$-function by Stanislav Sykora (2016) at his web-space.

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