Grand orbit relations in wandering domains

TL;DR

This paper explores the nature of grand orbit relations in wandering domains of transcendental entire maps, providing criteria to distinguish between discrete and indiscrete types and revealing their coexistence within a single domain.

Contribution

It introduces new criteria for classifying grand orbit relations and demonstrates their coexistence in wandering domains, supported by a novel quasiconformal surgery technique.

Findings

Discrete and indiscrete grand orbit relations can coexist in a wandering domain.

Criteria to distinguish between discrete and indiscrete relations are established.

A new quasiconformal surgery method is developed for analysis.

Abstract

One of the fundamental distinctions in McMullen and Sullivan's description of the Teichm\"uller space of a complex dynamical system is between discrete and indiscrete grand orbit relations. We investigate these on the Fatou set of transcendental entire maps and provide criteria to distinguish between the two types. Furthermore, we show that discrete and indiscrete grand orbit relations may coexist non-trivially in a wandering domain, a phenomenon which does not occur for any other type of Fatou component. One of the tools used is a novel quasiconformal surgery technique of independent interest.