# Countably and entropy expansive homeomorphisms with the shadowing property

## TL;DR

This paper investigates the properties of countably and entropy expansive homeomorphisms with the shadowing property, revealing conditions under which they are expansive and relating these concepts to Anosov diffeomorphisms.

## Contribution

It establishes new links between countable and entropy expansivity, and characterizes when such homeomorphisms are conjugate to Anosov systems.

## Key findings

- Countably expansive homeomorphisms with shadowing are expansive on transitive points.
- Examples of countably expansive homeomorphisms with infinite chain-recurrent classes are provided.
- For surface homeomorphisms with shadowing and full non-wandering set, countable and entropy cw-expansivity are equivalent to being Anosov.

## Abstract

We discuss the dynamics beyond topological hyperbolicity considering homeomorphisms satisfying the shadowing property and generalizations of expansivity. It is proved that transitive countably expansive homeomorphisms satisfying the shadowing property are expansive in the set of transitive points. This is in contrast with pseudo-Anosov diffeomorphisms of the two-dimensional sphere that are transitive, cw-expansive, satisfy the shadowing property but the dynamical ball in each transitive point contains a Cantor subset. We exhibit examples of countably expansive homeomorphisms that are not finite expansive, satisfy the shadowing property and admits an infinite number of chain-recurrent classes. We further explore the relation between countable and entropy expansivity and prove that for surface homeomorphisms $f:S\to S$ satisfying the shadowing property and $\Omega(f)=S$, both countably expansive and entropy cw-expansive are equivalent to being topologically conjugate to an Anosov diffeomorphism.