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  "body": "\n## Statement\n\nLet $A = (\\R / \\Z) \\times [0, 1]$ be the closed [[annulus|0CW52NV5TC]], with the area measure $d\\theta \\, dr$ given by [[Lebesgue measure|2Z2QVWZ262]].\nLet $f \\colon A \\to A$ be a [[homeomorphism|0QT00PB5G2]] such that:\n\n1. $f$ preserves area.\n2. $f$ maps each boundary circle $(\\R / \\Z) \\times \\set{0}$ and $(\\R / \\Z) \\times \\set{1}$ to itself.\n3. $f$ twists the boundary circles in opposite directions: some lift $F \\colon \\R \\times [0, 1] \\to \\R \\times [0, 1]$ of $f$ to the [[universal cover|Y1YPE6PM52]], written $F(x, r) = (x + g(x, r), h(x, r))$, satisfies $g(x, 0) \u003c 0 \u003c g(x, 1)$ for all $x \\in \\R$.\n\nThen $f$ has at least two fixed points.\n\n## Remarks\n\nPoincaré stated the theorem shortly before his death and checked many special cases, and Birkhoff published a proof in 1913.\nBirkhoff's proof works with the lift $F$ on the strip $\\R \\times [0, 1]$.\nThe twist condition makes the horizontal displacement $g$ change sign between the two boundaries.\nArea preservation then rules out the ways in which $F$ could still avoid a fixed point.\nApplied to iterates $f^q$ and lifts shifted by integers $p$, the theorem gives periodic orbits of every rotation number $p/q$ between the rotation numbers of the two boundaries.\nArea preservation is necessary: the map with lift $F(x, r) = (x + r - 1/2, r^2)$ satisfies 2 and 3, but a fixed point would need $r^2 = r$ and $r - 1/2 \\in \\Z$, which is impossible.\nThis map is not area preserving, since its Jacobian determinant is $2r$.\n\n## Explanations\n\n### L1 human\n\nPicture a ring that is twisted so that its inner edge turns one way and its outer edge turns the other way.\nIf the twist does not squeeze or stretch any area, then at least two points of the ring end up exactly where they started.\nThe theorem is a basic tool for finding periodic motions in mechanics.\n\n### L2 human\n\nArea-preserving twist maps of an annulus arise as return maps of Hamiltonian systems with two degrees of freedom, for example near an elliptic periodic orbit or in billiards.\nThe theorem then produces infinitely many periodic orbits, one family for each rational rotation number in the twist interval.\nIt is a forerunner of symplectic topology: Arnold's conjecture on fixed points of Hamiltonian diffeomorphisms was proposed as a higher-dimensional generalisation.\n",
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