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 diffeomorphisms were introduced by Dmitri Victorovich Anosov, who proved that their behaviour was in an appropriate sense generic (when they exist at all).
[1] Three closely related definitions must be distinguished: 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.
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 2023[update] has no answer for dimension over 3.
A sufficient condition for transitivity is that all points are nonwandering:
This in turn holds for codimension-one Anosov diffeomorphisms (i.e., those for which the contracting or the expanding subbundle is one-dimensional)[2] and for codimension one Anosov flows on manifolds of dimension greater than three[3] as well as Anosov flows whose Mather spectrum is contained in two sufficiently thin annuli.
volume-preserving Anosov diffeomorphism is ergodic.
there exists a unique SRB measure (the acronym stands for Sinai, Ruelle and Bowen)
is of full volume, where As an example, this section develops the case of the Anosov flow on the tangent bundle of a Riemann surface of negative curvature.
This flow can be understood in terms of the flow on the tangent bundle of the Poincaré half-plane model of hyperbolic geometry.
Riemann surfaces of negative curvature may be defined as Fuchsian models, that is, as the quotients of the upper half-plane and a Fuchsian group.
For the following, let H be the upper half-plane; let Γ be a Fuchsian group; let M = H/Γ be a Riemann surface of negative curvature as the quotient of "M" by the action of the group Γ, and let
be the tangent bundle of unit-length vectors on the manifold M, and let
The Lie algebra of PSL(2,R) is sl(2,R), and is represented by the matrices which have the algebra The exponential maps define right-invariant flows on the manifold of
The connection to the Anosov flow comes from the realization that
Lie vector fields being (by definition) left invariant under the action of a group element, one has that these fields are left invariant under the specific elements
In other words, the spaces TP and TQ are split into three one-dimensional spaces, or subbundles, each of which are invariant under the geodesic flow.
The final step is to notice that vector fields in one subbundle expand (and expand exponentially), those in another are unchanged, and those in a third shrink (and do so exponentially).
More precisely, the tangent bundle TQ may be written as the direct sum or, at a point
, the direct sum corresponding to the Lie algebra generators Y, J and X, respectively, carried, by the left action of group element g, from the origin e to the point q.
These spaces are each subbundles, and are preserved (are invariant) under the action of the geodesic flow; that is, under the action of group elements
shrinks exponentially as exp(-t) under the action of
This may be seen by examining how the group elements commute.
The geodesic flow is invariant, but the other two shrink and expand: and where we recall that a tangent vector in
corresponds to a geodesic on the upper half plane, passing through the point
The action is the standard Möbius transformation action of SL(2,R) on the upper half-plane, so that A general geodesic is given by with a, b, c and d real, with
Horocycles correspond to the motion of the normal vectors of a horosphere on the upper half-plane.