Multiply connected wandering domains of meromorphic functions: internal dynamics and connectivity

TL;DR

This paper extends the classification of wandering domains' internal dynamics from simply connected to multiply connected cases in meromorphic functions, revealing diverse behaviors influenced by connectivity and providing new examples and convergence insights.

Contribution

It generalizes Benini et al.'s classification scheme to multiply connected wandering domains and constructs a meromorphic function with a wandering domain lacking eventual connectivity.

Findings

Not all pairs of points in multiply connected wandering domains behave similarly hyperbolically.

Connectivity significantly influences the internal dynamics of wandering domains.

A meromorphic function with a wandering domain without eventual connectivity is constructed.

Abstract

We discuss how the nine-way classification scheme devised by Benini et al. for the dynamics of simply connected wandering domains of entire functions, based on the long-term behaviour of the hyperbolic distance between iterates of pairs of points and also the distance between orbits and the domains' boundaries, carries over to the general case of multiply connected wandering domains of meromorphic functions. Most strikingly, we see that not all pairs of points in such a wandering domain behave in the same way relative to the hyperbolic distance, and that the connectivity of the wandering domain greatly influences its possible internal dynamics. After illustrating our results with the well-studied case of Baker wandering domains, we further illustrate the diversity of multiply connected wandering domains in general by constructing a meromorphic function with a wandering domain without eventual connectivity. Finally, we show that an analogue of the "convergence to the boundary" classification of Benini et al. does hold in general, and add new information about how this convergence takes place.