Das, Tarun, Keonhee Lee, David Richeson, and Jim Wiseman. Spectral Decomposition for Topologically Anosov Homeomorphisms on Noncompact and Non-Metrizable Spaces. Topology and its Applications 160, no. 1 (2013): 149-158. https://www.sciencedirect.com/science/article/pii/S0166864112004270

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We discuss 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 Smaleʼs spectral decomposition theorem to topologically Anosov homeomorphisms on first countable, locally compact, paracompact, Hausdorff spaces.

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