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  "body": "\n## Summary\n\nTopological entropy is a nonnegative number, possibly infinite, that measures the exponential growth rate of the number of distinguishable orbits of a continuous map.\nIt is an invariant of topological conjugacy, so it can show that two systems are not conjugate.\nPositive entropy is a common sign of complicated, chaotic behaviour.\nThrough the [[variational principle|HHR8HP3R6G]] it equals the supremum of the measure-theoretic entropies of the invariant measures.\n\n## Explanations\n\n### L1 human\n\nSuppose you can only see the state of a system up to a fixed small precision.\nCount how many different orbit segments of length $n$ you can tell apart.\nIf this number grows like $e^{hn}$, the topological entropy is $h$.\nFor a rotation, nearby points stay close forever, so the count does not grow and the entropy is $0$.\nFor the doubling map of the circle, the count doubles at each step, and the entropy is $\\log 2$.\n",
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