N-expansive homeomorphisms with the shadowing property

TL;DR

This paper explores the dynamics of n-expansive homeomorphisms with shadowing, demonstrating their properties, existence, and implications for topological mixing and expansiveness on compact metric spaces.

Contribution

It constructs examples of n-expansive homeomorphisms with shadowing that are not (n-1)-expansive, and analyzes their stable sets and limit shadowing properties.

Findings

Existence of n-expansive homeomorphisms with shadowing but not (n-1)-expansive

Presence of infinite chain-recurrent classes in such systems

Identification of limit shadowing properties in these dynamics

Abstract

We discuss the dynamics of $n$-expansive homeomorphisms with the shadowing property defined on compact metric spaces. For every $n\in\mathbb{N}$, we exhibit an $n$-expansive homeomorphism, which is not $(n-1)$-expansive, has the shadowing property and admits an infinite number of chain-recurrent classes. We discuss some properties of the local stable (unstable) sets of $n$-expansive homeomorphisms with the shadowing property and use them to prove that some types of the limit shadowing property are present. This deals some direction to the problem of non-existence of topologically mixing $n$-expansive homeomorphisms that are not expansive.