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  "body": "\n## Statement\n\nLet $f$ be an orientation-preserving [[homeomorphism|0QT00PB5G2]] of the circle $S^1 = \\R / \\Z$, with [[rotation number|DP4WWQBJJF]] $\\rho = \\rho(f)$.\n\n1. Let $\\rho = p/q$ with $p, q$ coprime integers and $q \\ge 1$. Then $f$ has a [[periodic point|T75CZ5H47N]], and every periodic point of $f$ has least period $q$. The omega-limit set of every point is a periodic orbit.\n2. Let $\\rho$ be irrational. Then $f$ has no periodic point. There is a continuous surjective map $h \\colon S^1 \\to S^1$ of degree one, weakly preserving the cyclic order, with $h \\circ f = R_\\rho \\circ h$. Here $R_\\rho(x) = x + \\rho$.\n3. If $\\rho$ is irrational and $f$ has a [[dense|0BSJSX7ME2]] orbit, then $h$ is a homeomorphism, so $f$ is [[topologically conjugate|HNXXAPQKG3]] to the rotation $R_\\rho$.\n\n## Remarks\n\nThe converse parts also hold: $f$ has a periodic point if and only if $\\rho$ is rational.\nFor part 1, one shows that $\\rho(F) = p/q$ for a lift $F$ forces the equation $F^q(x) = x + p$ to have a solution, which gives a periodic point.\nSince $f$ preserves the cyclic order, a periodic orbit of $f$ is arranged on the circle like an orbit of the rotation by $p/q$.\nThis forces its least period to be $q$.\nBetween two consecutive fixed points of $f^q$ the map $f^q$ moves every point in the same direction.\nSo every orbit of $f^q$ is monotone and converges to a fixed point of $f^q$.\nFor part 2, $h$ is built by sending the orbit of a point $x$ to the orbit of $0$ under $R_\\rho$.\nThis works because $f$ orders an orbit on the circle in the same way as $R_\\rho$ does, and $h$ then extends by monotonicity.\nWhen $\\rho$ is irrational and $f$ has no dense orbit, the omega-limit set of every point is the same [[Cantor set|0BXNDZFQME]], and $f$ is only [[semiconjugate|AEXJNRBASV]] to $R_\\rho$.\nExample: for the rotation $R_{2/5}$ every point has period $5$, and for an irrational $\\alpha$ the map $R_\\alpha$ is an [[irrational rotation|2Q1J7RZ4TK]] with $h = \\id$.\n\n## Explanations\n\n### L1 human\n\nA homeomorphism of the circle moves points around while keeping their order.\nPoincaré showed that the long-run average turn, the rotation number, tells what kind of motion is possible.\nIf it is a fraction $p/q$, there are points that come back after exactly $q$ steps, and all other points drift towards such cycles.\nIf it is irrational, nothing ever comes back exactly, and the map looks like a rigid irrational rotation, possibly after collapsing some intervals.\n\n### L2 human\n\nThe theorem reduces the topological study of circle homeomorphisms to one number plus a little extra structure.\nIn the irrational case the remaining question is whether the semiconjugacy $h$ is a conjugacy, that is, whether $f$ has wandering intervals.\nDenjoy's theorem answers this for smooth enough diffeomorphisms: a $C^2$ diffeomorphism with irrational rotation number is conjugate to the rotation.\nIn the rational case the periodic orbits all have the same period and the same combinatorics as the rotation by $p/q$.\nThis is the starting point of the study of Arnold tongues.\n",
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