Wandering dynamics of transcendental functions

TL;DR

This paper demonstrates that complex wandering and escaping dynamics of holomorphic functions can be realized by transcendental meromorphic functions, providing new insights into the structure of such dynamical systems and their relation to entire functions.

Contribution

It constructs transcendental meromorphic functions that replicate given holomorphic dynamics on compact sets, including wandering and oscillating behaviors, with applications to wandering domains.

Findings

Realization of holomorphic dynamics by transcendental functions

Construction of entire functions with prescribed wandering domains

Extension of results to oscillating dynamics

Abstract

We show that any uniformly escaping and wandering dynamics of a holomorphic function on a compact subset of the plane can be realised by a transcendental meromorphic function on $\mathbb{C}$. More precisely, let $\varphi$ be a holomorphic function on an open subset of the complex plane, and suppose that $K$ is a compact set such that $\varphi$ and all its iterates $\varphi^n$ are defined on $K$, and $\varphi^n(K)\to\infty$ as $n\to\infty$. We prove that there exist a transcendental meromorphic function $f\colon\mathbb{C}\to\widehat{\mathbb{C}}$ and a compact set $\widetilde{K}$ such that the dynamics of $f$ on the orbit of $\widetilde{K}$ is conjugate, via a smooth change of coordinate close to the identity, to that of $\varphi$ on the orbit of $K$. If $K$ does not separate the plane, the function $f$ may be chosen to be entire. If all iterates of $\varphi$ are univalent on $K$, we can take $\widetilde{K}=K$. We also prove a similar theorem for oscillating dynamics. Finally, we use our results to answer a number of questions of Benini et al. concerning wandering domains of entire functions.