---
tag: MPFD24TYW2
kind: concept
name: Hyperbolic set
msc: [37D05, 37D20]
sources:
  - url: https://en.wikipedia.org/wiki/Hyperbolic_set
same_as:
  wikidata: Q3042004
  wikipedia: Hyperbolic set
---

## Summary

A hyperbolic set of a diffeomorphism is a compact invariant set with a splitting of the tangent space at each of its points.
One part is uniformly contracted and the other is uniformly expanded.
It extends the notion of a hyperbolic fixed point from one point to a whole set, possibly a Cantor set or the whole manifold.
Hyperbolic sets have stable and unstable manifolds, the shadowing property and robustness under perturbation.
They are the setting of uniformly hyperbolic dynamics, which includes [[Anosov diffeomorphisms|RCQH8P6DPQ]] and the [[Smale horseshoe|N63Z6EA6YS]].

## Explanations

### L1 human

Picture a map that, near some set of points, squeezes in one direction and stretches in another, at every point of the set and at a uniform rate.
Such a set is hyperbolic.
Nearby points drift apart along the stretched direction, so the motion on a large hyperbolic set can be chaotic, yet the clear geometry makes it possible to analyse.
The simplest case is a single saddle point, such as the origin for the map $(x, y) \mapsto (x/2, 2y)$.
See the [[definition|NY9MCZXY6N]].