---
tag: RCQH8P6DPQ
kind: concept
name: Anosov diffeomorphism
msc: [37D20]
specializes: [116TPXR5FS]
sources:
  - url: https://en.wikipedia.org/wiki/Anosov_diffeomorphism
same_as:
  wikidata: Q56297274
  wikipedia: Anosov diffeomorphism
---

## Summary

An Anosov diffeomorphism is a diffeomorphism of a compact manifold for which the whole manifold is a [[hyperbolic set|MPFD24TYW2]].
At every point it contracts some directions and expands the complementary ones, uniformly.
Anosov diffeomorphisms are [[structurally stable|D4MGFAHVMT]], satisfy [[Axiom A|Z26Z4XZFVA]] and admit [[Markov partitions|4VNHVDHTNN]].
The basic examples are the hyperbolic toral automorphisms, such as [[Arnold's cat map|BSCZN8Y1AN]].

## Explanations

### L1 human

Take a torus and a map that stretches it in one direction and squeezes it in another, everywhere and by a fixed rate.
Such a map mixes points very fast, like dough being kneaded, but it never tears the surface.
Arnold's cat map, given by the matrix with rows $(2, 1)$ and $(1, 1)$ acting on the torus, is the standard example.
Small smooth changes of an Anosov diffeomorphism do not change its orbit structure.
See the [[definition|X1P48KQFXQ]].