An Anosov diffeomorphism is a C^1 diffeomorphism ϕ of a manifold M to itself such that the tangent bundle of M is hyperbolic with respect to ϕ. Very few classes of Anosov diffeomorphisms are known. The best known is Arnold's cat map. A hyperbolic linear map R^n->R^n with integer entries in the transformation matrix and determinant ± 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.
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