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  "body": "\n## Example\n\nLet $Q = [0,1]^2$, and let $H_0 = [0,1] \\times [0, \\tfrac13]$ and $H_1 = [0,1] \\times [\\tfrac23, 1]$ be two horizontal strips.\nConsider a diffeomorphism $f$ of the plane that bends $Q$ into a horseshoe shape and is affine on the strips:\n$$f(x, y) = \\left(\\tfrac{x}{3}, \\; 3y\\right) \\text{ on } H_0, \\qquad f(x, y) = \\left(1 - \\tfrac{x}{3}, \\; 3 - 3y\\right) \\text{ on } H_1.$$\nThen $f(H_0) = [0, \\tfrac13] \\times [0,1]$ and $f(H_1) = [\\tfrac23, 1] \\times [0, 1]$, and the middle strip of $Q$ is mapped outside $Q$.\nLet $\\Lambda = \\bigcap_{n \\in \\Z} f^n(Q)$ be the set of points whose whole orbit stays in $Q$.\n\nCheck of the structure.\nConsider the maps $y \\mapsto 3y$ on $[0, \\tfrac13]$ and $y \\mapsto 3 - 3y$ on $[\\tfrac23, 1]$.\nA point stays in $Q$ for all forward times exactly when its $y$-coordinate never leaves $[0, \\tfrac13] \\cup [\\tfrac23, 1]$ under these maps.\nThis means that $y$ lies in the middle-thirds [[Cantor set|0BXNDZFQME]] $C$.\nBy the same argument for $f^{-1}$, a point stays in $Q$ for all backward times exactly when $x \\in C$.\nSo $\\Lambda = C \\times C$, a Cantor set.\nThe map $h \\colon \\Lambda \\to \\set{0,1}^{\\Z}$, with $h(p)_n = i$ when $f^n(p) \\in H_i$, is a homeomorphism with $h \\circ f = \\sigma \\circ h$, so $f|_\\Lambda$ is [[topologically conjugate|HNXXAPQKG3]] to the full shift on two symbols.\nOn $\\Lambda$, $Df$ equals $\\operatorname{diag}(\\tfrac13, 3)$ or $\\operatorname{diag}(-\\tfrac13, -3)$.\nSo the horizontal direction is contracted and the vertical one expanded by the factor $3$, and $\\Lambda$ is a [[hyperbolic set|MPFD24TYW2]] with $C = 1$ and $\\lambda = \\tfrac13$.\n\n## Remarks\n\nThrough the conjugacy, $f|_\\Lambda$ has $2^n$ points of period dividing $n$, a dense orbit, and topological entropy $\\log 2$.\nHyperbolicity makes the horseshoe robust: every diffeomorphism $C^1$-close to $f$ has an invariant Cantor set on which it is conjugate to the same shift.\nSmale showed that a transverse homoclinic point of a diffeomorphism produces a horseshoe for some iterate, which explains why such points imply chaotic behaviour.\n\n## Explanations\n\n### L1 human\n\nTake a square, stretch it to a long thin strip, bend the strip into a horseshoe, and lay it back across the square.\nSome points leave the square, and the points that stay forever form a dust of points, a Cantor set.\nEach such point is described by an infinite list of $0$s and $1$s, telling whether it is in the lower or the upper part of the square at each time.\nEvery list occurs, so the motion is as unpredictable as tossing a coin, yet it comes from a simple smooth map.\n",
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