Anosov Diffeomorphism

An Anosov diffeomorphism is a $C^1$ Diffeomorphism $\phi$ such that the Manifold $M$ is Hyperbolic with respect to $\phi$. Very few classes of Anosov diffeomorphisms are known. The best known is Arnold's Cat Map.

A Hyperbolic linear map $\Bbb{R}^n\to\Bbb{R}^n$ with Integer entries in the transformation Matrix and Determinant $\pm 1$ is an Anosov diffeomorphism of the $n$-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 $\phi:M\to M$ is an Anosov diffeomorphism on a Compact Riemannian Manifold and the Nonwandering Set $\Omega(\phi)$ of $\phi$ is $M$, then $\phi$ 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 $n$-Torus is Topologically Conjugate to an Anosov Automorphism, and also that Anosov diffeomorphisms are $C^1$ Structurally Stable.

See also Anosov Automorphism, Axiom A Diffeomorphism, Dynamical System

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.