Anosov river

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

definition

A flow on a Riemannian manifold is called Anosov flow if it is a continuous, -invariant decomposition ${\displaystyle \phi ^{t}}$ ${\displaystyle M}$ ${\displaystyle \phi ^{t}}$

${\displaystyle TM=E^{c}\oplus E^{s}\oplus E^{u}}$

the tangent is such that is tangent to the flow direction and respectively by be contracted or expanded uniformly, d. H. there is with ${\displaystyle TM}$ ${\displaystyle E^{c}}$ ${\displaystyle E^{s}}$ ${\displaystyle E^{u}}$ ${\displaystyle D\phi ^{t}}$ ${\displaystyle C\geq 1,\lambda >0}$

${\displaystyle \parallel D\phi ^{t}(v^{s})\parallel \leq Ce^{-\lambda t}\parallel v^{s}\parallel \ \forall v^{s}\in E^{s},t>0}$

${\displaystyle \parallel D\phi ^{t}(v^{u})\parallel \geq {\frac {1}{C}}e^{\lambda t}\parallel v^{u}\parallel \ \forall v^{u}\in E^{u},t>0}$.

The sub-bundles and are called stable and unstable bundles, the direct sums and are called weakly stable and weakly unstable bundles. ${\displaystyle E^{s}}$ ${\displaystyle E^{u}}$ ${\displaystyle E^{ss}:=E^{c}\oplus E^{s}}$ ${\displaystyle E^{uu}:=E^{c}\oplus E^{u}}$

Differentiability of the distributions

In general, the distributions and are only continuous and not necessarily differentiable. Benoist-Foulon-Labourie have shown that the stable and unstable bundle of an Anosov flow on a compact manifold of negative section curvature are only bundles if (except for -reparametrization) it is the geodetic flow of a locally symmetric space . ${\displaystyle E^{s}}$ ${\displaystyle E^{u}}$ ${\displaystyle C^{\infty }}$ ${\displaystyle C^{\infty }}$

Integral manifolds

The sub-bundles and are integrable , their integral manifolds are called weakly stable and weakly unstable manifolds. The weakly stable or weakly unstable manifolds of an Anosov river each form a tight foliage . ${\displaystyle E^{c}\oplus E^{s}}$ ${\displaystyle E^{c}\oplus E^{u}}$

Analogously, the integral manifolds of or are referred to as stable or unstable manifolds. ${\displaystyle E^{s}}$ ${\displaystyle E^{u}}$

Examples

• The geodetic flow (on the unit tangential bundle ) of a Riemann manifold of negative sectional curvature is an Anosov flow, its stable and unstable manifolds are unit tangential bundles of horospheres .
• The suspension flow of an Anosov diffeomorphism , for example a hyperbolic automorphism of the torus , is an Anosov flow.

properties

• Periodic orbites are close together .
• A measure-maintaining Anosov flow is ergodic .

literature

Stephen Smale : Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 1967 747-817 pdf

supporting documents

1. ^ Yves Benoist , Patrick Foulon , François Laborie : Flots d'Anosov à distributions stable et instable différentiables. J. Amer. Math. Soc. 5, no. 1, 33-74 (1992). pdf ( Memento of the original from October 23, 2005 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. 
2. ↑ Joseph Plante : Anosov flows. Amer. J. Math. 94: 729-754 (1972). pdf
3. ^ Gustav Hedlund : The dynamics of geodesic flows. Bull. Amer. Math. Soc. 45 (1939), no. 4, 241-260. pdf
4. ↑ Dmitri Anosov : Geodesic flows on closed Riemann manifolds with negative curvature. Proceedings of the Steklov Institute of Mathematics, No. 90 (1967). Translated from the Russian by S. Feder American Mathematical Society, Providence, RI 196