Anosov Diffeomorphism

An Anosov diffeomorphism is a  diffeomorphism  of a manifold  to itself such that the tangent bundle of  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 of Negative Curvature." Trudy Mat. Inst. Steklov 90, 1-209, 1970.

Smale, S. "Differentiable Dynamical Systems." Bull. Amer. Math. Soc. 73, 747-817, 1967.