Local product structure for expansive homeomorphisms

TL;DR

This paper proves that expansive homeomorphisms with dense hyperbolic periodic points on compact manifolds exhibit a local product structure, which becomes uniform under certain conditions, leading to conjugacy with linear Anosov diffeomorphisms.

Contribution

It establishes the existence of a local product structure for expansive homeomorphisms and shows it is uniform when certain hyperbolic periodic points are present, connecting to Anosov systems.

Findings

Local product structure exists on an open dense subset of the manifold.

The local product structure is uniform if a hyperbolic periodic point has codimension one.

Homeomorphisms with these properties are conjugate to linear Anosov diffeomorphisms.

Abstract

Let $f\colon M\to M$ be an expansive homeomorphism with dense topologically hyperbolic periodic points, $M$ a compact manifold. Then there is a local product structure in an open and dense subset of $M$. Moreover, if some topologically hyperbolic periodic point has codimension one, then this local product structure is uniform. In particular, we conclude that the homeomorphism is conjugated to a linear Anosov diffeomorphism of a torus.