Dynamical systems and ergodic theory Definition

Anosov flow

Builds on Smooth dynamical system Vector field Flow of a vector field Compact space

Definition

Let $M$ be a compact Riemannian manifold with norm $\norm{\cdot}$ on each tangent space. Let $X$ be a $C^1$ vector field on $M$ with no zeros, and let $\varphi_t$ be its flow. The flow $\varphi_t$ is an Anosov flow if the following hold.

For each $x \in M$ there is a splitting $T_x M = E^s_x \oplus E^0_x \oplus E^u_x$ , where $E^0_x$ is the line spanned by $X(x)$ .
The splitting is invariant: $D\varphi_t(E^s_x) = E^s_{\varphi_t(x)}$ and $D\varphi_t(E^u_x) = E^u_{\varphi_t(x)}$ for all $t \in \R$ .
There are constants $C > 0$ and $\lambda > 0$ such that, for all $x \in M$ and $t \ge 0$ , $\norm{D\varphi_t v} \le C e^{-\lambda t}\norm{v}$ for $v \in E^s_x$ and $\norm{D\varphi_{-t} v} \le C e^{-\lambda t}\norm{v}$ for $v \in E^u_x$ .

Remarks

Example: the geodesic flow on the unit tangent bundle of a compact surface of constant curvature $-1$ is an Anosov flow. In the model $\Gamma \backslash \mathrm{PSL}_2(\R)$ , the flow is right multiplication by $a_t = \operatorname{diag}(e^{t/2}, e^{-t/2})$ , and $E^s$ , $E^u$ are tangent to the orbits of the two horocycle flows. Conjugating $\begin{pmatrix} 1 & s \\ 0 & 1\end{pmatrix}$ by $a_t$ multiplies $s$ by $e^{t}$ or by $e^{-t}$ , which gives clause 3 with $\lambda = 1$ . Another example is the suspension flow of an Anosov diffeomorphism with constant roof function. Non-example: the linear flow $\varphi_t(x) = x + t v$ on the torus $\R^n/\Z^n$ has derivative the identity, so no direction is contracted and clause 3 fails.

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Anosov flow definition

Tag	`EN4A9MM7RP`
URI	`m2://EN4A9MM7RP`
Kind	definition
Writing	W0 stub
MSC	37D20, 37D40
Defines	Anosov flow
Uses	Smooth dynamical system, Vector field, Flow of a vector field, Compact space

Anosov flow

Definition

Remarks

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