Equality of the dynamical sets of two commuting transcendental entire functions

TL;DR

This paper proves that for certain pairs of commuting transcendental entire functions, key dynamical sets such as escaping sets and Julia sets are identical, extending previous results and considering more general relations.

Contribution

The paper establishes the equality of dynamical sets for commuting transcendental entire functions of specific forms, extending prior work and generalizing to functions satisfying polynomial relations.

Findings

Escaping sets of f and g coincide

Julia sets of f and g are identical

Results extend previous theorems to broader classes of functions

Abstract

In this paper, we study the dynamics of commuting transcendental entire functions $f$ and $g$, where $g$ is of the form $af^p + b$ with $a,b \in \C$, $p \in \N$, and $a \neq 0,1$. We establish that the escaping sets, filled Julia sets, and bungee sets of $f$ and $g$ all coincide. As an immediate consequence, we obtain in particular that the Julia sets of $f$ and $g$ are identical. Our theorem extends the 1998 result of Poon and Yang. Furthermore, following Wang and Yang, we consider a non-constant polynomial $Q$ and permutable entire functions $f$ and $g$ satisfying the relation $Q(g)=aQ(f)+b$, where $a(\neq 0,1), b \in \C$. In this more general setting, we also prove that the escaping sets, filled Julia sets, and bungee sets of $f$ and $g$ are equal.