Comments

• tfisher (2017-07-07 09:06:04 -0700 -0700):

  For surfaces it was shown by [Hiraide90] and [Lewowicz89] 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.

• tfisher (2017-07-07 09:41:17 -0700 -0700):

  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 [Vieitez96] . 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 [ABP09] .