Classifying simply connected wandering domains

TL;DR

This paper provides a comprehensive classification of the dynamics within simply connected wandering domains of transcendental entire functions, introducing new techniques for constructing and analyzing such domains.

Contribution

It offers a detailed classification framework based on hyperbolic distances and boundary behavior, along with a novel construction method for bounded, simply connected wandering domains.

Findings

Nine types of simply connected wandering domains are realizable.

New results on non-autonomous holomorphic dynamical systems.

A general technique for constructing wandering domains with prescribed dynamics.

Abstract

While the dynamics of transcendental entire functions in periodic Fatou components and in multiply connected wandering domains are well understood, the dynamics in simply connected wandering domains have so far eluded classification. We give a detailed classification of the dynamics in such wandering domains in terms of the hyperbolic distances between iterates and also in terms of the behaviour of orbits in relation to the boundaries of the wandering domains. In establishing these classifications, we obtain new results of wider interest concerning non-autonomous forward dynamical systems of holomorphic self maps of the unit disk. We also develop a new general technique for constructing examples of bounded, simply connected wandering domains with prescribed internal dynamics, and a criterion to ensure that the resulting boundaries are Jordan curves. Using this technique, based on approximation theory, we show that all of the nine possible types of simply connected wandering domain resulting from our classifications are indeed realizable.