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  "body": "\n## Summary\n\nThe rotation number of an orientation-preserving [[homeomorphism|0QT00PB5G2]] of the [[circle|060W5CBZP6]] is the average amount by which the map turns points around the circle, per step.\nFor the rotation by an angle $\\alpha$ it is $\\alpha$.\nIt is a topological invariant: homeomorphisms conjugate by an orientation-preserving homeomorphism have the same rotation number.\nPoincaré introduced it, and showed that it decides much of the dynamics: rational rotation number means periodic orbits, irrational rotation number means none.\nIt is the basic invariant of circle dynamics and of quasi-periodic motion.\n\n## Explanations\n\n### L1 human\n\nFollow a point as a map moves it around a circle many times, and count the total number of turns made.\nDivide by the number of steps; in the long run this ratio settles to a number, the rotation number.\nFor a clock hand that moves a quarter turn each tick, the rotation number is $1/4$.\nA map can move different points by different amounts, but the long-run average turn is the same for every starting point.\n",
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