---
tag: 3S543TYT8C
kind: definition
name: Markov partition
msc: [37D20, 37B10]
defines: 4VNHVDHTNN
uses: [MPFD24TYW2, 0Z1Y7JHGBR]
sources:
  - url: https://en.wikipedia.org/wiki/Markov_partition
---

## Definition

Let $M$ be a compact Riemannian manifold with distance $d$, and let $f \colon M \to M$ be a $C^1$ [[diffeomorphism|0Z1Y7JHGBR]].
Let $\Lambda \subseteq M$ be a [[hyperbolic set|MPFD24TYW2]] of $f$.
For $\varepsilon > 0$ and $x \in M$, let $W^s_\varepsilon(x) = \set{y \in M : d(f^n x, f^n y) \le \varepsilon \text{ for all } n \ge 0}$ and $W^u_\varepsilon(x) = \set{y \in M : d(f^{-n} x, f^{-n} y) \le \varepsilon \text{ for all } n \ge 0}$.
Assume that $\Lambda$ has a local product structure, with constants $\varepsilon > 0$ and $\delta > 0$.
This means that for all $x, y \in \Lambda$ with $d(x, y) < \delta$, the set $W^s_\varepsilon(x) \cap W^u_\varepsilon(y)$ is a single point $[x, y]$ of $\Lambda$.
A *rectangle* is a nonempty set $R \subseteq \Lambda$ of diameter less than $\delta$ such that $[x, y] \in R$ for all $x, y \in R$.
It is *proper* if $R$ is the closure of its interior, taken relative to $\Lambda$.
For $x \in R$ put $W^s(x, R) = W^s_\varepsilon(x) \cap R$ and $W^u(x, R) = W^u_\varepsilon(x) \cap R$.
A *Markov partition* of $\Lambda$ is a finite family $R_1, \ldots, R_k$ of proper rectangles such that:

1. $\Lambda = R_1 \cup \cdots \cup R_k$.
2. The interiors of $R_i$ and $R_j$ are disjoint for $i \ne j$.
3. If $x$ lies in the interior of $R_i$ and $f(x)$ lies in the interior of $R_j$, then $f(W^s(x, R_i)) \subseteq W^s(f(x), R_j)$ and $f(W^u(x, R_i)) \supseteq W^u(f(x), R_j)$.

## Remarks

Clause 3 says that $f$ maps stable slices into stable slices and stretches unstable slices across the whole rectangle.
Then the transition matrix $T_{ij} = 1$ when $f(\operatorname{int} R_i) \cap \operatorname{int} R_j \ne \emptyset$, and $T_{ij} = 0$ otherwise, defines a [[subshift of finite type|624XDXX8FN]] $\Sigma_T$.
The map $\pi(x) = \bigcap_{n \in \Z} f^{-n}(R_{x_n})$ is a continuous surjection $\Sigma_T \to \Lambda$ with $\pi \circ \sigma = f \circ \pi$.
Example: let $\Lambda$ be the invariant Cantor set of the [[Smale horseshoe|N63Z6EA6YS]], identified with $\set{0,1}^{\Z}$ so that $f$ becomes the shift.
Fix $m \ge 1$, and for each word $w$ of length $2m + 1$ let $R_w$ be the set of points whose coordinates $-m, \ldots, m$ spell $w$.
For large $m$ these sets are small.
The point $[x, y]$ takes its coordinates of index $n \ge 0$ from $x$ and of index $n < 0$ from $y$, so each $R_w$ is a rectangle.
It is open and closed, so it is proper.
Applying the shift moves the agreeing future of a stable slice to the future of $f(x)$, which gives clause 3.
Non-example: removing one set $R_w$ from this family leaves a family that violates clause 1.
Sinai constructed Markov partitions for Anosov diffeomorphisms, and Bowen for the basic sets of Axiom A diffeomorphisms.