Dynamic rays of bounded-type transcendental self-maps of the punctured plane

TL;DR

This paper investigates the structure of escaping points in a class of transcendental self-maps of the punctured plane, revealing that these points form complex, Cantor bouquet-like structures connecting essential singularities.

Contribution

It establishes the existence of curves connecting escaping points to singularities and describes the detailed topological structure of the escaping set for these functions.

Findings

Escaping points lie in the Julia set.

Every escaping point can be connected to an essential singularity by a uniform escaping curve.

The escaping set contains Cantor bouquets accumulating at 0 and infinity.

Abstract

We study the escaping set of functions in the class $\mathcal B^*$, that is, holomorphic functions $f:\mathbb C^*\to\mathbb C^*$ for which both zero and infinity are essential singularities, and the set of singular values of $f$ is contained in a compact annulus of $\mathbb C^*$. For functions in the class $\mathcal B^*$, escaping points lie in their Julia set. If $f$ is a composition of finite order transcendental self-maps of $\mathbb C^*$ (and hence, in the class $\mathcal B^*$), then we show that every escaping point of $f$ can be connected to one of the essential singularities by a curve of points that escape uniformly. Moreover, for every essential itinerary $e\in\{0,\infty\}^\mathbb N$, we show that the escaping set of $f$ contains a Cantor bouquet of curves that accumulate to $\{0,\infty\}$ according to $e$ under iteration by $f$.