Two remarks on transcendental shift-like maps on $\mathbb{C}^N$

TL;DR

This paper investigates the dynamics of transcendental shift-like maps in complex N-dimensional space, establishing the non-emptiness of their Julia sets and providing an example with an escaping wandering domain, highlighting differences from polynomial cases.

Contribution

It proves the non-emptiness of Julia sets for transcendental shift-like maps and presents an example with an escaping wandering domain, contrasting polynomial shift map dynamics.

Findings

Julia set of transcendental shift-like maps is non-empty

Provided an example with an escaping wandering domain

Highlighted contrast with polynomial shift map dynamics

Abstract

In \cite{Bedford}, the dynamics of a particular polynomial diffeomorphism of $\mathbb{C}^N$, called a polynomial shift-like map of type $\nu$, has been studied as a higher dimensional analog of H\'enon maps. In this note, we prove that the Julia set of their transcendental counterpart is non-empty. In addition, an example of a transcendental shift-like map with an escaping wandering domain has been provided which, in fact, showcases a contrast with the dynamics of a polynomial shift map.