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

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.

Expansive homeomorphisms on non-Hausdorff spaces

“…So, if the neighborhood U of the diagonal of Proposition 2.4 can be taken as a closed neighborhood then we conclude by Theorem 2.7 that X is metric. Consequently, the topological expansivity defined in [3] is metric expansivity if the space is compact.…”

Topological stability of a sequence of maps on a compact metric space

“…In [4,5], authors have generalized results of [5,18] for symplectomorphisms. Recently in [8], authors have studied expansiveness, shadowing, topological stability and decomposition theorems for homeomorphisms on non-compact and nonmetrizable spaces.…”

Topological stability and spectral decomposition for homeomorphisms on noncompact spaces

“…Aoki [1] extended the result to homeomorphisms on compact metric spaces as follows: if f is an expansive homeomorphism with the shadowing property on a compact metric space, then Ω(f ) can be written as a finite union of disjoint closed invariant sets on which f is topologically transitive. Afterwards, there are many works that generalize the spectral decomposition theorem to multidimentional dynamical systems (e.g., see [8]), and to homeomorphisms on noncompact spaces (e.g., see [5]). Recently, Cordeiro et.…”