The definition of an Anosov map is the same as for an Anosov diffeomorphism except that instead of being a diffeomorphism, it is a map. In particular, an Anosov map is a C^1 map f of a manifold M to itself such that the tangent bundle of M is hyperbolic with respect to f. A trivial example is to map all of M to a single point of M. Here, the eigenvalues are all zero. A less trivial example is an expanding map on the circle S^1, e.g., x↦2x (mod 1), where S^1 is identified with the real numbers (mod 1). Here, all the eigenvalues equal 2 (i.e., the eigenvalue at each point of S^1). Note that this map is not a diffeomorphism because f(x + (1/2)) = f(x), so it has no inverse.

Anosov diffeomorphism | Anosov flow | Arnold's cat map