Anosov flow - Knowlpedia

A $C^1$ flow $\varphi_t:N\to N$ on a compact manifold is an Anosov flow if there is a continuous invariant splitting

TN=E^s\oplus E^0\oplus E^u,

where the generating vector field $X(x)=\left.\frac{d}{dt}\right|_{t=0}\varphi_t(x)$ is nowhere zero and $E^0_x=\operatorname{span}\{X(x)\}$ , and constants $C,\lambda>0$ satisfy

\|D\varphi_t v^s\|\le Ce^{-\lambda t}\|v^s\|, \qquad \|D\varphi_{-t}v^u\|\le Ce^{-\lambda t}\|v^u\|

for all $t\ge0$ , $v^s\in E^s$ , and $v^u\in E^u$ .

Invariance

The splitting is invariant: $D\varphi_t(E^\bullet_x)=E^\bullet_{\varphi_t(x)}$ . Vectors in $E^s$ contract forward in time, while vectors in $E^u$ contract backward in time.

Principal example

The geodesic flow on the unit tangent bundle of a compact negatively curved manifold is Anosov. Its hyperbolic splitting supports symbolic descriptions and stable/unstable fractal sets.

References

Dmitry Anosov, Geodesic Flows on Closed Riemannian Manifolds with Negative Curvature, AMS, 1969. DOI record.