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  "body": "\n## Definition\n\nLet $M$ be a compact Riemannian manifold with distance $d$, and let $f \\colon M \\to M$ be a $C^1$ [[diffeomorphism|0Z1Y7JHGBR]].\nLet $\\Lambda \\subseteq M$ be a [[hyperbolic set|MPFD24TYW2]] of $f$.\nFor $\\varepsilon \u003e 0$ 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 $W^u_\\varepsilon(x) = \\set{y \\in M : d(f^{-n} x, f^{-n} y) \\le \\varepsilon \\text{ for all } n \\ge 0}$.\nAssume that $\\Lambda$ has a local product structure, with constants $\\varepsilon \u003e 0$ and $\\delta \u003e 0$.\nThis means that for all $x, y \\in \\Lambda$ with $d(x, y) \u003c \\delta$, the set $W^s_\\varepsilon(x) \\cap W^u_\\varepsilon(y)$ is a single point $[x, y]$ of $\\Lambda$.\nA *rectangle* is a nonempty set $R \\subseteq \\Lambda$ of diameter less than $\\delta$ such that $[x, y] \\in R$ for all $x, y \\in R$.\nIt is *proper* if $R$ is the closure of its interior, taken relative to $\\Lambda$.\nFor $x \\in R$ put $W^s(x, R) = W^s_\\varepsilon(x) \\cap R$ and $W^u(x, R) = W^u_\\varepsilon(x) \\cap R$.\nA *Markov partition* of $\\Lambda$ is a finite family $R_1, \\ldots, R_k$ of proper rectangles such that:\n\n1. $\\Lambda = R_1 \\cup \\cdots \\cup R_k$.\n2. The interiors of $R_i$ and $R_j$ are disjoint for $i \\ne j$.\n3. 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 $f(W^u(x, R_i)) \\supseteq W^u(f(x), R_j)$.\n\n## Remarks\n\nClause 3 says that $f$ maps stable slices into stable slices and stretches unstable slices across the whole rectangle.\nThen 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]] $\\Sigma_T$.\nThe map $\\pi(x) = \\bigcap_{n \\in \\Z} f^{-n}(R_{x_n})$ is a continuous surjection $\\Sigma_T \\to \\Lambda$ with $\\pi \\circ \\sigma = f \\circ \\pi$.\nExample: let $\\Lambda$ be the invariant Cantor set of the [[Smale horseshoe|N63Z6EA6YS]], identified with $\\set{0,1}^{\\Z}$ so that $f$ becomes the shift.\nFix $m \\ge 1$, and for each word $w$ of length $2m + 1$ let $R_w$ be the set of points whose coordinates $-m, \\ldots, m$ spell $w$.\nFor large $m$ these sets are small.\nThe point $[x, y]$ takes its coordinates of index $n \\ge 0$ from $x$ and of index $n \u003c 0$ from $y$, so each $R_w$ is a rectangle.\nIt is open and closed, so it is proper.\nApplying the shift moves the agreeing future of a stable slice to the future of $f(x)$, which gives clause 3.\nNon-example: removing one set $R_w$ from this family leaves a family that violates clause 1.\nSinai constructed Markov partitions for Anosov diffeomorphisms, and Bowen for the basic sets of Axiom A diffeomorphisms.\n",
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