Problem 80

For surfaces it was shown by [1] and [2] 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.

References

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 [1]. 

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 [2].

References