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  "body": "\n## Statement\n\nLet $X$ be a [[compact|0TVJZYB6SH]] [[metric space|0VKD1TP7F3]] and let $f \\colon X \\to X$ be a [[continuous map|0VGFHTM75Y]].\nLet $\\mathcal{M}(f)$ be the set of $f$-[[invariant|B232EZ1FMB]] Borel [[probability measures|059TJN0NZ1]] on $X$.\nFor $\\mu \\in \\mathcal{M}(f)$, let $h_\\mu(f)$ be the [[Kolmogorov–Sinai entropy|X1D1C7VH6P]] of $f$ with respect to $\\mu$.\nThen the [[topological entropy|BDZYBXBFDE]] satisfies\n$$h_{\\mathrm{top}}(f) = \\sup_{\\mu \\in \\mathcal{M}(f)} h_\\mu(f).$$\nThe supremum may be taken over the ergodic measures in $\\mathcal{M}(f)$ only.\n\n## Remarks\n\nThe set $\\mathcal{M}(f)$ is nonempty by the [[Krylov–Bogolyubov theorem|1DHZ81S0YE]].\nThe inequality $h_\\mu(f) \\le h_{\\mathrm{top}}(f)$ follows by comparing a finite measurable partition with an open cover, and Misiurewicz gave a short proof of the reverse inequality.\nFor the reverse inequality one builds, from large $(n, \\varepsilon)$-separated sets, averaged measures whose weak-$\\ast$ limit is invariant and has entropy close to $h_{\\mathrm{top}}(f)$.\nThe restriction to ergodic measures follows from the ergodic decomposition, since $h_\\mu$ is the average of the entropies of the ergodic components.\nExample: for the full shift on $k$ symbols, $h_{\\mathrm{top}} = \\log k$, and the Bernoulli measure giving each symbol mass $1/k$ has entropy $\\log k$, so the supremum is attained.\nThe supremum is not always attained; a measure that attains it is called a measure of maximal entropy.\n\n## Explanations\n\n### L1 human\n\nThere are two ways to measure how chaotic a system is.\nTopological entropy counts distinguishable orbits, while measure-theoretic entropy measures the unpredictability of typical orbits for a given statistics.\nThe variational principle says that the first is the best possible value of the second.\n\n### L2 human\n\nThe theorem lets one compute topological entropy from a well-chosen invariant measure, and conversely bounds all measure-theoretic entropies by a topological quantity.\nIt is the starting point of the thermodynamic formalism, where entropy is replaced by pressure and the maximizing measures become equilibrium states.\nMeasures of maximal entropy exist when $\\mu \\mapsto h_\\mu(f)$ is upper semicontinuous, for example for expansive homeomorphisms.\n",
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