## Anosov Diffeomorphism

An Anosov diffeomorphism is a Diffeomorphism such that the Manifold is Hyperbolic with respect to . Very few classes of Anosov diffeomorphisms are known. The best known is Arnold's Cat Map.

A Hyperbolic linear map with Integer entries in the transformation Matrix and Determinant is an Anosov diffeomorphism of the -Torus. Not every Manifold admits an Anosov diffeomorphism. Anosov diffeomorphisms are Expansive, and there are no Anosov diffeomorphisms on the Circle.

It is conjectured that if is an Anosov diffeomorphism on a Compact Riemannian Manifold and the Nonwandering Set of is , then is Topologically Conjugate to a Finite-to-One Factor of an Anosov Automorphism of a Nilmanifold. It has been proved that any Anosov diffeomorphism on the -Torus is Topologically Conjugate to an Anosov Automorphism, and also that Anosov diffeomorphisms are Structurally Stable.

Anosov, D. V. Geodesic Flow on Closed Riemannian Manifolds with Negative Curvature.'' Proc. Steklov Inst., A. M. S. 1969.
Smale, S. Differentiable Dynamical Systems.'' Bull. Amer. Math. Soc. 73, 747-817, 1967.