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  "body": "\n## Definition\n\nLet $S$ be a closed orientable [[surface|VGJJVBZ1MQ]].\nA *singular measured foliation* on $S$ is a [[foliation|98HTB7MSPX]] $\\mathcal{F}$ of $S \\setminus \\Sigma$, for a finite set $\\Sigma$, with a transverse measure $\\mu$.\nHere $\\mu$ is a measure on arcs transverse to the leaves, invariant under sliding along the leaves.\nNear each point of $\\Sigma$ the foliation must be a $p$-prong singularity for some $p \\ge 3$.\nA [[homeomorphism|0QT00PB5G2]] $f \\colon S \\to S$ is *pseudo-Anosov* if there are $\\lambda \u003e 1$ and singular measured foliations $(\\mathcal{F}^s, \\mu^s)$, $(\\mathcal{F}^u, \\mu^u)$ on $S$ such that:\n\n1. $\\mathcal{F}^s$ and $\\mathcal{F}^u$ have the same singular set $\\Sigma$ and are transverse away from $\\Sigma$.\n2. $f$ maps the leaves of $\\mathcal{F}^s$ to leaves of $\\mathcal{F}^s$, and $f_* \\mu^s = \\lambda^{-1} \\mu^s$.\n3. $f$ maps the leaves of $\\mathcal{F}^u$ to leaves of $\\mathcal{F}^u$, and $f_* \\mu^u = \\lambda \\mu^u$.\n\nThe number $\\lambda$ is the *stretch factor* (or dilatation) of $f$.\n\n## Remarks\n\nExample: on the torus $\\R^2 / \\Z^2$, the map induced by $A = \\begin{pmatrix} 2 \u0026 1 \\\\ 1 \u0026 1 \\end{pmatrix}$, [[Arnold's cat map|BSCZN8Y1AN]], is pseudo-Anosov with no singular points.\nThe eigenvalues of $A$ are $\\lambda = \\frac{3 + \\sqrt 5}{2} \u003e 1$ and $\\lambda^{-1}$.\nThe foliations by lines parallel to the two eigenvectors, with transverse measures given by distance across the lines, are contracted by $\\lambda^{-1}$ and stretched by $\\lambda$.\n\nNon-example: the [[Dehn twist|W890NRHGS4]] induced by $\\begin{pmatrix} 1 \u0026 1 \\\\ 0 \u0026 1 \\end{pmatrix}$ on the torus has only the eigenvalue $1$.\nIt preserves the horizontal foliation without stretching it, so no $\\lambda \u003e 1$ fits clauses 2 and 3.\n\nOn a closed surface of genus $g \\ge 2$, the Euler–Poincaré formula forces singular points, since the foliations cannot be nonsingular.\nFor surfaces with punctures, $1$-prong singularities are also allowed at the punctures.\nThe stretch factor $\\lambda$ is an algebraic integer, and $\\log \\lambda$ is the topological entropy of $f$, the least entropy among all homeomorphisms isotopic to $f$.\n",
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