---
tag: 4VNHVDHTNN
kind: concept
name: Markov partition
msc: [37D20, 37B10]
sources:
  - url: https://en.wikipedia.org/wiki/Markov_partition
same_as:
  wikidata: Q15978248
  wikipedia: Markov partition
---

## Summary

A Markov partition of a hyperbolic system is a cover of the space by finitely many pieces, called rectangles.
The map moves these rectangles in a very regular way.
Each rectangle is a product of a stable and an unstable piece, and the map stretches the unstable pieces across whole rectangles.
Recording which rectangle a point visits at each time turns the system into a [[subshift of finite type|624XDXX8FN]].
In this way Markov partitions make [[symbolic dynamics|EWFWTSVY0B]] a tool for [[Anosov diffeomorphisms|RCQH8P6DPQ]] and for the basic sets of [[Axiom A|Z26Z4XZFVA]] systems.

## Explanations

### L1 human

Cut the space into a few boxes, and name each box by a letter.
Follow a point under the map and write down the letter of the box it is in at each time.
For a Markov partition the boxes are chosen so well that the allowed strings of letters are exactly those that avoid a finite list of forbidden pairs.
So the complicated motion is described by a simple rule on sequences of letters.
For the [[Smale horseshoe|N63Z6EA6YS]], two boxes already suffice, and every sequence of two letters occurs.
See the [[definition|3S543TYT8C]].