Reddy examples of expansive maps. Related to Anosov diffeos. Are expansive diffeos likely to be Anosov?
For surfaces it was shown by  and  that there are no expansive homeomorphisms on the 2-sphere, that any expansive homeomorphism of the 2-torus is conjugate to an Anosov diffeomorphism, and for higher genus surfaces the expanisve homeomorphism is conjuage to a pseudo-Anosov homeomorphism.
For higher dimensions there are partial classifications. For expansive homeomorphisms of compact 3-manifolds with dense topologically hyperbolic periodic points the manifold is the 3-torus and the expansive homeomorphism is conjugate to an Anosov diffeomorphism .
For the general higher dimensional setting if an expansive homeomorphism has dense topologically hyperbolic periodic points, then there exists a local product structure for the homeomorphism on an open and dense set in the manifold. Furthermore, if a hyperbolic periodic point for the expansive homeomorphism is codimension-one, then the expansive homeomorphism is conjuage to a hyperbolic toral automorphism .