Spectral decomposition for topologically Anosov homeomorphisms on noncompact and non-metrizable spaces

TL;DR

This paper extends the concepts of expansivity, shadowing, and spectral decomposition to topologically Anosov homeomorphisms on noncompact, non-metrizable spaces, broadening classical results from compact metric spaces.

Contribution

It introduces topological definitions for key dynamical properties and generalizes Smale's spectral decomposition theorem to a wider class of spaces.

Findings

Generalized spectral decomposition theorem for topologically Anosov homeomorphisms

Extended definitions of expansivity, shadowing, and chain recurrence to non-metric spaces

Proved theorems analogous to classical results in a broader topological setting

Abstract

We introduce topological definitions of expansivity, shadowing, and chain recurrence for homeomorphisms. They generalize the usual definitions for metric spaces. We prove various theorems about topologically Anosov homeomorphisms (maps that are expansive and have the shadowing property) on noncompact and non-metrizable spaces that generalize theorems for such homeomorphisms on compact metric spaces. The main result is a generalization of Smale's spectral decomposition theorem to topologically Anosov homeomorphisms on first countable locally compact paracompact Hausdorff spaces.