Abstract
In this paper, we introduce the notions of expansiveness, shadowing property and topological stability for homeomorphisms on noncompact metric spaces which are dynamical properties and equivalent to the classical definitions in case of compact metric spaces. Then we extend the Walters's stability theorem and Smale's spectral decomposition theorem to homeomorphisms on locally compact metric spaces.
Highlights
Let X be a compact metrizable space and f be a homeomorphism of X onto itself
Chung and Lee [3] extended the notion of topological stability from homeomorphisms to group actions on compact metric spaces, and proved that if an action of a finite generated group action is expansive and has the shadowing property it is topologically stable
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, Ω(f ) can be written as a finite union of disjoint closed invariant sets on which f is topologically transitive
Summary
Let X be a compact metrizable space and f be a homeomorphism of X onto itself. Fix any metric d for X (throughout the paper, this term means that d is a metric compatible with the topology on X). Chung and Lee [3] extended the notion of topological stability from homeomorphisms to group actions on compact metric spaces, and proved that if an action of a finite generated group action is expansive and has the shadowing property it is topologically stable. 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, Ω(f ) can be written as a finite union of disjoint closed invariant sets on which f is topologically transitive. We introduce the notions of expansiveness, shadowing property and topological stability for homeomorphisms on noncompact metric spaces which are dynamical properties and equivalent to the classical definitions in case of compact metric spaces.
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