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  "body": "\n## Definition\n\nLet $(X, d)$ be a [[metric space|0VKD1TP7F3]] and let $f \\colon X \\to X$ be a [[homeomorphism|0QT00PB5G2]].\nThe map $f$ is *expansive* if there is a constant $c \u003e 0$ such that for all $x, y \\in X$ with $x \\neq y$ there is an integer $n \\in \\Z$ with $d(f^n(x), f^n(y)) \u003e c$.\nSuch a $c$ is an *expansivity constant* of $f$.\n\n## Remarks\n\nOn a compact metric space, expansiveness does not depend on the choice of a metric that gives the topology, although the constant does.\nExample: the two-sided full [[shift|A1XE59ZHNM]] $\\sigma$ on $\\set{0, 1}^{\\Z}$ with $d(x, y) = 2^{-\\min\\set{\\abs{i} : x_i \\neq y_i}}$ and $c = 1/2$.\nIf $x \\neq y$, pick $i$ with $x_i \\neq y_i$; then $\\sigma^i(x)$ and $\\sigma^i(y)$ differ at coordinate $0$, so their distance is $1 \u003e 1/2$.\nNon-example: an [[isometry|02HPP5VCZH]] such as the rotation $x \\mapsto x + \\alpha \\bmod 1$ of the circle.\nFor any $c \u003e 0$, two distinct points at distance less than $c$ stay at that distance for all $n \\in \\Z$.\nAn expansive homeomorphism of a compact metric space has finite [[topological entropy|BDZYBXBFDE]].\n\n## Explanations\n\n### L1 human\n\nA homeomorphism is expansive when any two distinct points are pushed at least a fixed distance apart at some time, past or future.\n",
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