Anosov diffeomorphism

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:
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 "C"1 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: formula_1.
Also, it is unknown if every formula_2 volume-preserving Anosov diffeomorphism is ergodic. Anosov proved it under a formula_3 assumption. It is also true for formula_4 volume-preserving Anosov diffeomorphisms.
For formula_3 transitive Anosov diffeomorphism formula_6 there exists a unique SRB measure (the acronym stands for Sinai, Ruelle and Bowen) formula_7 supported on formula_8 such that its basin formula_9 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 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 formula_11 be the tangent bundle of unit-length vectors on the manifold "M", and let formula_12 be the tangent bundle of unit-length vectors on "H". Note that a bundle of unit-length vectors on a surface is the principal bundle of a complex line bundle.
Lie vector fields.
One starts by noting that formula_12 is isomorphic to the Lie group PSL(2,R). This group is the group of orientation-preserving isometries of the upper half-plane. 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 formula_17, and likewise on formula_18. Defining formula_19 and formula_20, these flows define vector fields on "P" and "Q", whose vectors lie in "TP" and "TQ". These are just the standard, ordinary Lie vector fields on the manifold of a Lie group, and the presentation above is a standard exposition of a Lie vector field.
Anosov flow.
The connection to the Anosov flow comes from the realization that formula_21 is the geodesic flow on "P" and "Q". 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 formula_21 of the geodesic flow. 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 formula_24, 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". That is, one has formula_26 and formula_27. 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 formula_28.
To compare the lengths of vectors in formula_29 at different points "q", one needs a metric. Any inner product at formula_30 extends to a left-invariant Riemannian metric on "P", and thus to a Riemannian metric on "Q". The length of a vector formula_31 expands exponentially as exp(t) under the action of formula_21. The length of a vector formula_33 shrinks exponentially as exp(-t) under the action of formula_21. Vectors in formula_35 are unchanged. 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 formula_39 is given by the derivative, with respect to "t", of the curve formula_40, the setting formula_41.
Geometric interpretation of the Anosov flow.
When acting on the point formula_42 of the upper half-plane, formula_21 corresponds to a geodesic on the upper half plane, passing through the point formula_42. 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 formula_47. The curves formula_48 and formula_40 are called horocycles. Horocycles correspond to the motion of the normal vectors of a horosphere on the upper half-plane.