Let $z$ be a complex number. Meromorphic here means meromorphic on all of the complex plane $C$. Lets define a fractal $A_f$ on the complex plane as the result of iterating a meromorphic function $f(z)$ an infinite amount of times and then coloring according to the value. Thus $A_f(z)=f(f(f(...f(z)...)))$ and the color is decided upon the value (or limit cycle or divergence) of $A(z)$. It is required that $A_f$ has a fractal structure and not just e.g. $A_f(z)=0$ for all $z$. As example $f(z)=z^2+1$ gives the famous Mandelbrot fractal.

Let $k$ be a nonzero complex number. Does there exist a meromorphic function $f(z)$ for every $k$ such that $f(z)$ is not periodic but $A_f(z)=A_f(z+k)$ is always true and $A_f(z)$ is not double periodic ?

How to solve this problem ?


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