Countably and entropy expansive homeomorphisms with the shadowing property

We discuss the dynamics beyond topological hyperbolicity considering homeomorphisms satisfying shadowing property and generalizations of expansivity. It is proved that transitive countably expansive are in set points. This contrast with pseudo-Anosov diffeomorphisms two-dimensional sphere transitive, cw-expansive, satisfy but dynamical ball each point contains a Cantor subset. exhibit examples not finite expansive, admits an infinite number chain-recurrent classes. further explore relation between countable entropy expansivity prove for surface f : S → stretchy="false">( stretchy="false">) = encoding="application/x-tex">\Omega (f)=S , both cw-expansive equivalent to being topologically conjugate Anosov diffeomorphism.