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  "body": "\n## Statement\n\nLet $S$ be a closed orientable [[surface|VGJJVBZ1MQ]], and let $f \\colon S \\to S$ be an orientation-preserving [[homeomorphism|0QT00PB5G2]].\nThen $f$ is isotopic to a homeomorphism $g$ of at least one of the following three kinds:\n\n1. $g$ is periodic: $g^n = \\id_S$ for some $n \\ge 1$.\n2. $g$ is reducible: $g$ maps to itself a nonempty finite union $C$ of pairwise disjoint simple closed curves on $S$. No curve of $C$ bounds a disc, and no two are isotopic.\n3. $g$ is [[pseudo-Anosov|GYW7GYABD9]].\n\nA pseudo-Anosov homeomorphism is neither periodic nor reducible, and it is not isotopic to a homeomorphism of the first two kinds.\n\n## Remarks\n\nThe statement is about the elements of the [[mapping class group|JV5NYG913V]] of $S$: each mapping class is periodic, reducible or pseudo-Anosov, and the third kind excludes the first two.\nA mapping class can be both periodic and reducible.\nThe theorem also holds for compact surfaces with boundary and for surfaces with finitely many punctures.\n\nExample: on the torus, mapping classes correspond to matrices $A$ in the [[special linear group|0V9PMN0RBH]] $SL(2, \\Z)$.\nIf $\\abs{\\tr A} \u003c 2$, then $A$ has finite order and the class is periodic.\nIf $\\abs{\\tr A} = 2$, then $A$ has an integer eigenvector, and the class preserves the corresponding simple closed curve.\nFor $A \\neq \\pm I$ it is a power of a [[Dehn twist|W890NRHGS4]] up to sign and is reducible.\nIf $\\abs{\\tr A} \u003e 2$, then $A$ has real eigenvalues $\\lambda^{\\pm 1}$ with $\\abs{\\lambda} \u003e 1$, and the class is Anosov, a pseudo-Anosov class without singularities.\n\nThurston's proof considers the action of $f$ on a compactification of the [[Teichmüller space|ZZGC6BZNJ5]] of $S$ by projective measured foliations.\nThis compactification is a closed ball, and the proof uses the Brouwer fixed-point theorem.\nA fixed point inside the ball gives a periodic class.\nA fixed point on the boundary gives either a reducing curve system or the stable and unstable foliations of a pseudo-Anosov map.\nBers gave another proof by minimizing a quasiconformal dilatation over the isotopy class.\n\n## Explanations\n\n### L1 human\n\nAny way of deforming a surface back onto itself can be straightened out into one of three basic types.\nIt may repeat itself after finitely many steps, like a rotation.\nIt may keep some curves in place, and so split the surface into simpler pieces.\nOr it stretches and squeezes the whole surface everywhere, mixing it thoroughly.\nThe third type, pseudo-Anosov, is the generic and most interesting one.\n\n### L2 human\n\nThe theorem is the surface analogue of the classification of $2 \\times 2$ integer matrices into elliptic, parabolic and hyperbolic ones.\nBy cutting along the reducing curves and repeating, every mapping class decomposes into periodic and pseudo-Anosov pieces on subsurfaces.\nThis decomposition controls the dynamics: the least topological entropy in an isotopy class is the largest $\\log \\lambda$ of its pseudo-Anosov pieces.\nBy Thurston's hyperbolization theorem for fibered 3-manifolds, if $S$ has genus at least $2$, the mapping torus of $f$ has a hyperbolic structure exactly when $f$ is pseudo-Anosov.\n",
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