Sensitivity, local stable/unstable sets and shadowing

TL;DR

This paper investigates the structure of local stable and unstable sets in sensitive homeomorphisms with shadowing, revealing they contain perfect subsets, and extends results on countably expansive systems under certain conditions.

Contribution

It proves local stable/unstable sets contain perfect subsets and generalizes prior results on countably expansive homeomorphisms with shadowing or L-shadowing properties.

Findings

Local stable/unstable sets contain perfect subsets.

Positively countably expansive homeomorphisms with shadowing are on countable spaces.

Generalization of previous results on expansive systems.

Abstract

In this paper we study local stable/unstable sets of sensitive homeomorphisms with the shadowing property defined on compact metric spaces. We prove that local stable/unstable sets always contain a compact and perfect subset of the space. As a corollary we generalize results in [6] and [11] proving that positively countably expansive homeomorphisms defined on compact metric spaces satisfying either transitivity and the shadowing property, or the L-shadowing property, can only be defined in countably spaces.