Classifying multiply connected wandering domains

TL;DR

This paper classifies the internal hyperbolic geometric dynamics of multiply connected wandering domains in meromorphic functions, revealing six distinct types and constructing examples of a new global phenomenon unique to meromorphic functions.

Contribution

It introduces a combined approach using injectivity radii and hyperbolic distortions to fully describe and classify the internal dynamics of wandering domains, unifying previous methods.

Findings

Six types of internal hyperbolic dynamics identified.

Construction of a meromorphic function with a locally but not globally isometric wandering domain.

Global phenomena differ between entire and meromorphic functions.

Abstract

We study the internal dynamics of multiply connected wandering domains of meromorphic functions. We do so by considering the sequence of injectivity radii along the orbit of a base point, together with the hyperbolic distortions along the same orbit. The latter sequence had been previously used in this context; the former introduces geometric information about the shape of the wandering domains that interacts with the dynamic information given by the hyperbolic distortions. Using this idea, we complete the description of the internal dynamics of any wandering domain of a meromorphic function, and also unify previous approaches to the question. We conclude that the internal dynamics of a wandering domain, from the point of view of hyperbolic geometry, can be classified into six different types. Five of these types are realised by wandering domains of entire functions, while the sixth can only arise for meromorphic functions: a locally but not globally eventually isometric wandering domain. We construct a meromorphic function with such a domain, demonstrating that this new phenomenon does in fact occur. Our results show that, on a local level, the dynamics of multiply connected wandering domains of meromorphic functions is similar to those of entire functions, although new global phenomena can arise for non-entire functions.