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  "body": "\n## Statement\n\nLet $f$ be an orientation-preserving $C^1$ [[diffeomorphism|0KZZ70ZHXP]] of the circle $S^1 = \\R / \\Z$ such that $\\log f'$ has [[bounded variation|A8251BHCMP]]; for example, let $f$ be of class $C^2$.\nAssume that the [[rotation number|DP4WWQBJJF]] $\\rho$ of $f$ is irrational.\nThen $f$ is [[topologically conjugate|HNXXAPQKG3]] to the rotation $R_\\rho(x) = x + \\rho$: there is a homeomorphism $h$ of $S^1$ with $h \\circ f = R_\\rho \\circ h$.\n\n## Remarks\n\nBy Poincaré's classification, $f$ is semiconjugate to $R_\\rho$, and it is conjugate unless $f$ has a wandering interval, an open interval $J$ whose images $f^n(J)$, $n \\in \\Z$, are pairwise disjoint.\nThe proof rules out wandering intervals.\nLet $q_n$ be the denominators of the continued fraction of $\\rho$.\nA combinatorial argument finds intervals whose images under $f^{q_n}$ overlap little, and the Denjoy inequality bounds $\\abs{\\log (f^{q_n})'}$ by the total variation of $\\log f'$.\nFor a wandering interval $J$ this gives a constant $c \u003e 0$ with $\\abs{f^{q_n}(J)} + \\abs{f^{-q_n}(J)} \\ge c \\abs{J}$ for all $n$.\nBut the images of $J$ are pairwise disjoint, so their lengths tend to $0$, which is a contradiction.\n\nThe smoothness hypothesis cannot be dropped.\nDenjoy also built $C^1$ diffeomorphisms with irrational rotation number that have a wandering interval.\nFor them the omega-limit set of every point is a [[Cantor set|0BXNDZFQME]] and $f$ is not conjugate to a rotation.\nSuch examples exist even of class $C^{1 + \\alpha}$ for every $\\alpha \u003c 1$.\n\nThe conjugacy $h$ is only a homeomorphism in general; whether it is smooth depends on arithmetic properties of $\\rho$, a question studied by Arnold, Herman and Yoccoz.\n\n## Explanations\n\n### L1 human\n\nA smooth map of the circle that never brings a point back to itself turns out to be a rigid irrational rotation in disguise.\nOne can change coordinates on the circle, continuously, so that the map becomes a plain rotation.\nFor maps that are only once differentiable this can fail: some intervals may wander around the circle forever without overlapping.\n\n### L2 human\n\nThe theorem completes Poincaré's classification for smooth circle diffeomorphisms: up to topological conjugacy, a $C^2$ diffeomorphism with irrational rotation number is determined by that number.\nIt implies that every orbit of such a map is dense and that the map is minimal and [[uniquely ergodic|NZZE9KCAD8]], since the [[irrational rotation|2Q1J7RZ4TK]] has these properties.\nThe key tool, control of the distortion of $(f^n)'$ through bounded variation of $\\log f'$, reappears in many places in one-dimensional dynamics.\nDenjoy published the theorem in 1932.\n",
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