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  "body": "\n## Definition\n\nLet $(X, d)$ be a [[compact|0TVJZYB6SH]] [[metric space|0VKD1TP7F3]] and let $f \\colon X \\to X$ be a [[continuous map|0VGFHTM75Y]].\nFor an integer $n \\ge 1$, define the metric $d_n(x, y) = \\max_{0 \\le i \u003c n} d(f^i(x), f^i(y))$ on $X$.\nFor $\\varepsilon \u003e 0$, a set $E \\subseteq X$ is *$(n, \\varepsilon)$-separated* if $d_n(x, y) \u003e \\varepsilon$ for all distinct $x, y \\in E$.\nLet $s(n, \\varepsilon)$ be the largest size of an $(n, \\varepsilon)$-separated set, which is finite by compactness.\nThe *topological entropy* of $f$ is\n$$h_{\\mathrm{top}}(f) = \\lim_{\\varepsilon \\to 0} \\limsup_{n \\to \\infty} \\frac{1}{n} \\log s(n, \\varepsilon) \\in [0, \\infty],$$\nwhere $\\log$ is the [[natural logarithm|04DXAFN2ES]].\n\n## Remarks\n\nThe limit in $\\varepsilon$ exists because the $\\limsup$ grows as $\\varepsilon$ decreases.\nThe value depends only on the topology, not on the metric.\nAdler, Konheim and McAndrew defined it on any compact space through the growth of joins of [[open covers|049PXYHW1T]], and this gives the same number.\nTopologically conjugate maps have equal entropy, and $h_{\\mathrm{top}}(f^k) = k \\, h_{\\mathrm{top}}(f)$ for $k \\ge 1$.\nExample: the full one-sided [[shift|A1XE59ZHNM]] $\\sigma$ on $\\set{0, \\ldots, k-1}^{\\N}$ with $d(x, y) = 2^{-\\min\\set{i : x_i \\neq y_i}}$.\nThen $d_n(x, y) \u003e 2^{-m}$ exactly when $x$ and $y$ differ at some coordinate $i \u003c n + m - 1$, so $s(n, 2^{-m}) = k^{n + m - 1}$ and $h_{\\mathrm{top}}(\\sigma) = \\log k$.\nA map with zero entropy: an isometry has $d_n = d$, so $s(n, \\varepsilon)$ does not depend on $n$ and $h_{\\mathrm{top}} = 0$.\n\n## Explanations\n\n### L1 human\n\nTopological entropy is the exponential growth rate of the number of orbit segments that can be told apart at a fixed small resolution.\n",
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