Problem 80

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Reddy examples of expansive maps. Related to Anosov diffeos. Are expansive diffeos likely to be Anosov?



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.


  1. [Hiraide90] Hiraide K..  1990.  Expansive diffeomorphisms of surfaces are pseudo-Anosov. OsakaJ. Math.. 27:117-162.
  2. [Lewowicz89] Lewowicz J..  1989.  Expansive Homeomorphisms of surfaces. Bol. Soc. Bras. Math.. 20:113-133.

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].


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