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.

**References**

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.

© 1996-9

1999-05-25