Anosov Diffeomorphism
An Anosov diffeomorphism is a 
Diffeomorphism 
such that the Manifold 
isHyperbolic with respect to . Very few classes of Anosov diffeomorphisms are known. The bestknown is Arnold's Cat Map.
A Hyperbolic linear map 
with Integer entries in the transformationMatrix and Determinant 
is an Anosov diffeomorphism of the 
-Torus. Not everyManifold admits an Anosov diffeomorphism. Anosov diffeomorphisms are Expansive, and there are noAnosov diffeomorphisms on the Circle.
It is conjectured that if 
is an Anosov diffeomorphism on a CompactRiemannian Manifold and the Nonwandering Set 
of is , then isTopologically 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 anAnosov 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.
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