Shadowing, expansiveness and hyperbolic homeomorphisms

The purpose of this paper is to complete results concerning the class H of expansive homeomorphisms having the pseudo orbits tracing property on a compact metric space. We show that hyperbolic homeomorphisms introduced by Mane in [8] are exactly those in the class H; then by the result of [12, 20] they form a class equal to the Smale space introduced by Ruelle in [18]. Next, assuming that the phase space is a smooth manifold, we show that a diffeomorphism is Anosov if and only if it is in the class H and is a lower semi-continuity point of the map which assigns to any diffeomorphism the supremum of its expansive constants (possibly zero). Then we discuss the behavior of the dynamical systems generated by homeomorphisms from H near their basic sets.