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.
Contents
- Overview
- Anosov flow on tangent bundles of Riemann surfaces
- Lie vector fields
- Anosov flow
- Geometric interpretation of the Anosov flow
- References
Anosov diffeomorphisms were introduced by D. V. 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 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 2012 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
For
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 T1M be the tangent bundle of unit-length vectors on the manifold M, and let T1H 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 T1H 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 T1H = PSL(2,R), and likewise on T1M. Defining P = T1H and Q = T1M, 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
More precisely, the tangent bundle TQ may be written as the direct sum
or, at a point
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
To compare the lengths of vectors in
but the other two shrink and expand:
and
where we recall that a tangent vector in
Geometric interpretation of the Anosov flow
When acting on the point z = i of the upper half-plane,
A general geodesic is given by
with a, b, c and d real, with ad − bc = 1. The curves