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. Let $\Lambda \subseteq M$ be a hyperbolic set 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 $f(W^u(x, R_i)) \supseteq W^u(f(x), R_j)$.