Wandering dynamics of transcendental functions

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.

Figures (4)

• Figure 1: The continua $(X_n)_{n=0}^\infty$ and model map $\varphi$ from Example \ref{['ex:rabbit']}. Here $K$ is Douady's rabbit (in light gray), and the domain $U$ (in dark gray) is one of the components of $\textup{int}(K)$. Note that $\varphi^{n+3}(U)=\varphi^n(U)+9$ for all $n\geq 0$.
• Figure 2: Diagram for the proof of Lemma \ref{['lem:theta-n']}.
• Figure 3: Diagram for the proof of Theorem \ref{['thm:triangle1']}. For $0\leq n\leq j$, the restriction of $\Theta_{n,j}$ to $X_{n,n}=X_n$, is the composition $\theta_{n,j-1}\circ\dots\circ\theta_{n,n}$, which maps $X_n$ to $X_{n,j}$. In particular, $\Theta_{j,j}=\operatorname{id}$ and $\Theta_{j,j+1}=\theta_{j,j}$.
• Figure 4: Diagram for the proof of Theorem \ref{['thm:injective']}. On $X_n$, the map $\Theta_{n,j}$ is the composition $\theta_{n,j-1}\circ \dots \circ \theta_{n,n}\circ \theta_n$. Note that $\Theta_{0,j}=\operatorname{id}$ for all $j\geq 0$.

Theorems & Definitions (56)

• Theorem 1.1: Realising dynamics by escaping wandering domains
• Theorem 1.2: Realising dynamics by oscillating wandering domains
• Remark 1.3
• Remark 1.4
• Theorem 1.5: The entire case
• Example 1.6
• Remark 1.7
• Theorem 1.8: Realising univalent dynamics
• Corollary 1.9
• Theorem 2.1: Extension of Runge's theorem