Shadowing properties of L-hyperbolic homeomorphisms

We introduce the notion of L-hyperbolic homeomorphisms on compact metric spaces as a strict generalization of Axiom A diffeomorphisms and prove that the notion is equivalent to expansive homeomorphisms having the shadowing property and to Ruelleʹs Smale spaces. Furthermore, for L-hyperbolic homeomorphisms, both the Lipschitz shadowing property and the average shadowing property are shown.