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.