---
tag: ZHKS81A7QN
kind: concept
name: Pesin theory
msc: ["37D25", "37C40"]
sources:
  - url: https://en.wikipedia.org/wiki/Lyapunov_exponent
---

## Summary

Pesin theory is the study of smooth dynamical systems that are hyperbolic only almost everywhere with respect to an invariant measure.
Its starting point is the [[Oseledets theorem|NJ1PDJ3S4R]], which gives [[Lyapunov exponents|NJ1FJSNK6A]] at almost every point.
When no exponent is zero, the theory builds local stable and unstable manifolds at almost every point.
Their size and the angles between them may vary from point to point.
It links the exponents with entropy, periodic orbits and the existence of physical measures.

## Explanations

### L1 human

In a uniformly hyperbolic system, such as [[Arnold's cat map|BSCZN8Y1AN]], every point sees the same rate of stretching and squeezing.
Most smooth systems are not that regular: the stretching is there on average, but it can be weak and slow to show up near some points.
Pesin theory says that average stretching is enough to recover much of the uniform picture, point by point, for almost every point.
For example, take a smooth map of a surface that preserves area and has one positive and one negative exponent at almost every point.
Then almost every point has a short curve through it that the map squeezes and another one that the map stretches.
The price is that the length of these curves and the angle between them may get small near bad points.