Anosov flow

In mathematics are Anosov flows, named after Dmitri Viktorovich Anossow, a well -understood example of chaotic dynamics. They show the one hand, all the typical effects of chaotic behavior, on the other hand, a mathematical treatment easily accessible.

Definition

A flow on a Riemannian manifold is called Anosov flow if there is a continuous, invariant decomposition

Of the tangent there, so that is tangent to the flow direction and or be contracted or expanded by uniformly, ie there with

The sub-bundles and hot stable and unstable bundles, direct sums and hot weakly stable or weakly unstable bundle.

Differentiability of the distributions

In general, the distributions and only continuous and not necessarily differentiable. Benoist Foulon - Labourie have proved that the stable and unstable bundles of Anosov River are only bundles on a compact manifold of negative sectional curvature if it is (up to reparametrization - ) is the geodesic flow of a locally symmetric space.

Integral manifolds

The sub-bundles and integrable, its integral manifolds are called weakly stable or weakly unstable manifold. The weak stable and weak unstable manifolds of Anosov River each form a taut foliation.

Be referred to the integral manifolds of stable or unstable manifold or as analog.

Examples

• The geodesic flow ( on the Einheitstangentialbündel ) a Riemannian manifold of negative sectional curvature is an Anosov flow, its stable and unstable manifold are Einheitstangentialbündel of Horosphären.
• The suspension flow a Anosov diffeomorphism, for example a hyperbolic automorphism of the torus, a Anosov flow.

Properties

• Periodic orbits are dense.
• A measure - preserving Anosov flow is ergodic.