---
tag: 8FQJXSA4DT
kind: concept
name: Anosov flow
msc: [37D20, 37D40]
specializes: [116TPXR5FS]
sources:
  - url: https://en.wikipedia.org/wiki/Anosov_diffeomorphism
---

## Summary

An Anosov flow is a flow on a compact manifold with a splitting of the tangent bundle into three parts.
They are the flow direction, a uniformly contracted direction and a uniformly expanded direction.
It is the continuous-time analogue of an [[Anosov diffeomorphism|RCQH8P6DPQ]].
The [[geodesic flow|VF43FTDA1H]] of a compact Riemannian manifold of negative sectional curvature is an Anosov flow, a fact proved by Anosov.
Anosov flows are structurally stable up to orbit equivalence, and those of class $C^2$ that preserve a smooth volume are ergodic.

## Explanations

### L1 human

Imagine particles moving along a family of curves that fill a closed space.
Nearby curves either approach each other exponentially fast or separate exponentially fast, except for the trivial shift along the curve itself.
This is an Anosov flow.
The motion of a free particle sliding on a surface shaped everywhere like a saddle is the classical example.
See the [[definition|EN4A9MM7RP]].