Permutable entire functions and multiply connected wandering domains

TL;DR

This paper investigates the conditions under which permutable transcendental entire functions share the same Julia set, focusing on multiply connected wandering domains and providing progress on a longstanding conjecture.

Contribution

It establishes that if neither function has a simply connected wandering domain in the fast escaping set, then their Julia sets are equal.

Findings

Julia sets of permutable functions are equal under specified conditions

Analysis of multiply connected wandering domains advances understanding of complex dynamics

Provides partial resolution to a longstanding conjecture

Abstract

Let $f$ and $g$ be permutable transcendental entire functions. We use a recent analysis of the dynamical behaviour in multiply connected wandering domains to make progress on the long standing conjecture that the Julia sets of $f$ and $g$ are equal; in particular, we show that $J(f)=J(g)$ provided that neither $f$ nor $g$ has a simply connected wandering domain in the fast escaping set.