Anosov Diffeomorphism - Overview

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 as of 2005 has no answer. The only known examples are infranil manifolds, and it is conjectured that they are the only ones.

Another open problem is whether every Anosov diffeomorphism is transitive. All known Anosov diffeomorphisms are transitive. A sufficient condition for transitivity is nonwandering: .

Also it is unknown if every volume preserving Anosov diffeomorphism is ergodic. Anosov proved it under assumption. It is also true for volume preserving Anosov diffeomorphisms.

For transitive Anosov diffeomorphism there exists a unique SRB measure (stand for Sinai, Ruelle and Bowen) supported on such that its basin is of full volume, where