Transcendental dynamics: wandering domains of meromorphic functions

Rodrigues Ferreira, Gustavo (2023). Transcendental dynamics: wandering domains of meromorphic functions. PhD thesis The Open University.

DOI: https://doi.org/10.21954/ou.ro.0001590e

Abstract

This thesis explores the iteration of transcendental meromorphic functions, i.e., functions that are meromorphic in the complex plane and admit no analytic extension to the Riemann sphere. We focus on problems that are in some way related to wandering domains, regions of normality whose iterates are all disjoint. The first problem that we explore is that of internal dynamics. By using hyperbolic geometry, recent work of Benini et al. classified the internal dynamics of simply connected wandering domains and showed that they can fit neatly into one of nine types, each with uniform internal dynamics. We show that, for multiply connected wandering domains of entire functions, this uniformity breaks down, and that we have geometrically defined laminations that determine the internal dynamics of the wandering domain.

We also tackle the internal dynamics of multiply connected wandering domains of non-entire meromorphic functions. We show that they are much more complex and diverse, and investigate conditions that can recover the uniformity observed in the simply connected case. To further illustrate the richness of the meromorphic case, we construct (using approximation theory) a meromorphic function with a wandering domain with no eventual connectivity and (using quasiconformal surgery) a meromorphic function with a semi-contracting infinitely connected wandering domain.

The second problem that we investigate is that of commuting functions – one of the oldest in complex dynamics. Recent work by Bergweiler and Hinkkanen, and later Benini, Rippon, and Stallard, show that it is related –in the entire case – to the presence of fast escaping wandering domains. In adapting their techniques to the meromorphic setting, we prove the strongest statement to date on the problem of commuting meromorphic functions and draw attention to ping-pong orbits and wandering domains – a “meromorphic-only” phenomenon that further complicates the problem.

In the course of the thesis, we explore other problems that are tangentially related to those mentioned above. The first is the existence of what we call ill-behaved annuli coverings, infinite-degree maps without asymptotic values between topologica annuli. The second is the problem of symmetries of the Julia set, which – for rational functions – is known to be related to the problem of commuting functions. We discuss a conjecture by Kisaka and offer partial results towards classifying the symmetries of Julia sets of transcendental entire functions.

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