Anosov Map

The definition of an Anosov map is the same as for an Anosov diffeomorphism except that instead of being a diffeomorphism, it is a map. In particular, an Anosov map is a  map f of a manifold  to itself such that the tangent bundle of  is hyperbolic with respect to .

A trivial example is to map all of  to a single point of . Here, the eigenvalues are all zero. A less trivial example is an expanding map on the circle , e.g., , where  is identified with the real numbers (mod 1). Here, all the eigenvalues equal 2 (i.e., the eigenvalue at each point of ). Note that this map is not a diffeomorphism because , so it has no inverse.

A nontrivial example is formed by taking Arnold's cat map on the 2-torus , and crossing it with an expanding map on  to form an Anosov map on the 3-torus , where  denotes the Cartesian product. In other words,

REFERENCES:

Anosov, D. "Roughness of Geodesic Flows on Compact Riemannian Manifolds of Negative Curvature." Dokl. Akad. Nauk SSSR 145, 707-709, 1962. English translation in Soviet Math. Dokl. 3, 1068-1069, 1962.

Anosov, D. "Ergodic Properties of Geodesic Flows on Closed Riemannian Manifolds of Negative Curvature." Dokl. Akad. Nauk SSSR 151, 1250-1252, 1963. English translated in Soviet Math. Dokl. 4, 1153-1156, 1963.

Lichtenberg, A. J. and Lieberman, M. A. Regular and Chaotic Dynamics, 2nd ed. New York: Springer-Verlag, pp. 305-307, 1992.

Sondow, J. "Fixed Points of Anosov Maps of Certain Manifolds." Proc. Amer. Math. Soc. 61, 381-384, 1976.