---
tag: Z26Z4XZFVA
kind: concept
name: Axiom A
msc: [37D20]
sources:
  - url: https://en.wikipedia.org/wiki/Axiom_A
same_as:
  wikidata: Q4059942
  wikipedia: Axiom A
---

## Summary

A diffeomorphism satisfies Axiom A if its [[non-wandering set|EY2W34JZA4]] is a [[hyperbolic set|MPFD24TYW2]] in which periodic points are dense.
Smale introduced the condition to single out a large class of systems whose dynamics can be described completely.
For such systems the non-wandering set splits into finitely many basic pieces by the [[spectral decomposition theorem|6MTVY6NYY7]].
Each piece is coded by a [[subshift of finite type|624XDXX8FN]].
[[Anosov diffeomorphisms|RCQH8P6DPQ]], [[Morse–Smale systems|7AMQ8GS78Y]] and the [[Smale horseshoe|N63Z6EA6YS]] satisfy Axiom A.

## Explanations

### L1 human

Most points of a dynamical system wander off and never come back near where they started.
The interesting dynamics happens on the set of points that keep returning, the non-wandering set.
Axiom A asks that on this set the system stretches and squeezes uniformly, and that periodic orbits are everywhere in it.
Then the long-term behaviour can be cut into finitely many pieces, each understood by symbolic codes.
See the [[definition|S31RTTJPDC]].