# A note on Anosov homeomorphisms

## TL;DR

This paper provides an elementary proof that $ ext{alpha}$-expansive homeomorphisms with the local product structure property exhibit the shadowing property, linking expansiveness to pseudo orbit behavior in topological dynamics.

## Contribution

It offers a simplified proof connecting $ ext{alpha}$-expansiveness and shadowing, emphasizing the role of the local product structure property.

## Key findings

- $ ext{alpha}$-expansive homeomorphisms with local product structure have shadowing
- Elementary proof approach simplifies understanding of shadowing in expansive systems
- Reformulation of expansiveness in terms of pseudo orbits enhances theoretical insights

## Abstract

For an $\alpha$-expansive homeomorphism of a compact space we give an elementary proof of the following well-known result in topological dynamics: A sufficient condition for the homeomorphism to have the shadowing property is that it has the $\alpha$-shadowing property for one-jump pseudo orbits (known as the local product structure property). The proof relies on a reformulation of the property of expansiveness in terms of the pseudo orbits of the system.