In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold M is a certain type of mapping, from M to itself, with rather clearly marked local directions of "expansion" and "contraction". Anosov systems are a special case of Axiom A systems. Anosov diffeomorphisms were introduced by Dmitri Victorovich Anosov, who proved that their behaviour was in an appropriate sense generic (when they exist at all). Overview Three closely related definitions must be distinguished: If a differentiable map f on M has a hyperbolic structure on the tangent bundle, then it is called an Anosov map. Examples include the Bernoulli map, and Arnold's cat map. If the map is a diffeomorphism, then it is called an Anosov diffeomorphism. If a flow on a manifold splits the tangent bundle into three invariant subbundles, with one subbundle that is exponentially contracting, and one that is exponentially expanding, and a third, non-expanding, non-contracting one-dimensional sub-bundle (spanned by the flow direction), then the flow is called an Anosov flow. A classical example of Anosov diffeomorphism is the Arnold's cat map. Anosov proved that Anosov diffeomorphisms are structurally stable and form an open subset of mappings (flows) with the C1 topology. Not every manifold admits an Anosov diffeomorphism; for example, there are no such diffeomorphisms on the sphere . The simplest examples of compact manifolds admitting them are the tori: they admit the so-called linear Anosov diffeomorphisms, which are isomorphisms having no eigenvalue of modulus 1. It was proved that any other Anosov diffeomorphism on a torus is topologically conjugate to one of this kind. The problem of classifying manifolds that admit Anosov diffeomorphisms turned out to be very difficult, and still has no answer. The only known examples are infranil manifolds, and it is conjectured that they are the only ones. A sufficient condition for transitivity is that all points are nonwandering: . Also, it is unknown if every volume-preserving Anosov diffeomorphism is ergodic. Anosov proved it under a assumption. It is also true for volume-preserving Anosov diffeomorphisms. For transitive Anosov diffeomorphism there exists a unique SRB measure (the acronym stands for Sinai, Ruelle and Bowen) supported on such that its basin is of full volume, where Anosov flow on (tangent bundles of) Riemann surfaces As an example, this section develops the case of the Anosov flow on the tangent bundle of a Riemann surface of