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  "body": "\n## Statement\n\nLet $g \\ge 1$ and let $k_1, \\dots, k_n \\ge 0$ be integers with $k_1 + \\cdots + k_n = 2g - 2$.\nLet $\\mathcal{H}_1(k_1, \\dots, k_n)$ be the stratum of [[translation surfaces|CEHHBBD6NT]] $(X, \\omega)$ of genus $g$ and area $\\int_X \\frac{i}{2} \\omega \\wedge \\bar\\omega = 1$ whose form $\\omega$ has $n$ zeros of orders $k_1, \\dots, k_n$.\nLet $\\nu$ be the Masur–Veech measure on it, and let $g_t$, $t \\in \\R$, be the Teichmüller geodesic flow, the action of the diagonal matrices $\\begin{pmatrix} e^t \u0026 0 \\\\ 0 \u0026 e^{-t} \\end{pmatrix}$.\nLet $\\mathcal{C}$ be a connected component of $\\mathcal{H}_1(k_1, \\dots, k_n)$.\nThen:\n\n1. The total measure $\\nu(\\mathcal{C})$ is finite.\n2. The flow $g_t$ on $\\mathcal{C}$ is ergodic with respect to $\\nu$.\n\n## Remarks\n\nThe stratum $\\mathcal{H}(k_1, \\dots, k_n)$ of all such surfaces, of any area, has local period coordinates.\nTake a basis $\\gamma_1, \\dots, \\gamma_{2g + n - 1}$ of the relative homology $H_1(X, \\Sigma; \\Z)$, where $\\Sigma$ is the set of zeros.\nThen the map $(X, \\omega) \\mapsto (\\int_{\\gamma_j} \\omega)_j \\in \\C^{2g + n - 1}$ is a local chart.\nThe Masur–Veech measure on $\\mathcal{H}$ is Lebesgue measure in these charts, normalized so that the lattice of periods in $\\Z + i\\Z$ has covolume $1$.\nThe measure $\\nu$ on $\\mathcal{H}_1$ is the cone measure: $\\nu(E)$ is the Lebesgue measure of the set of $r \\cdot (X, \\omega)$ with $(X, \\omega) \\in E$ and $0 \u003c r \\le 1$.\nThe group $SL(2, \\R)$ acts on translation surfaces by acting linearly on the polygons, it preserves area, and the measure $\\nu$ is invariant under it.\n\nMasur and Veech proved the finiteness of the volume and the main consequence below independently in 1982, and the strata are not compact, so finiteness is not automatic.\nConsequence: for $\\nu$-almost every translation surface, the vertical straight-line flow is [[uniquely ergodic|NZZE9KCAD8]].\nThrough the first return map to a transversal, this gives Keane's conjecture: almost every [[interval exchange transformation|19JFF3GY9F]] with an irreducible permutation is uniquely ergodic.\n\n## Explanations\n\n### L1 human\n\nTranslation surfaces of a given shape type form a space, and this space has a natural notion of volume.\nThe theorem says that this volume is finite, and that a natural flow on the space, which stretches each surface horizontally and squeezes it vertically, mixes up the space thoroughly.\nFrom this one learns that for a randomly chosen surface, a straight line in a fixed direction spreads out evenly over the whole surface.\n\n### L2 human\n\nThe strata $\\mathcal{H}_1(k_1, \\dots, k_n)$ are moduli spaces of abelian differentials, fibred over the [[moduli space|FJZRW9P0K3]] of Riemann surfaces of genus $g$.\nThe flow $g_t$ projects to geodesics for the Teichmüller metric on [[Teichmüller space|ZZGC6BZNJ5]].\nErgodicity of $g_t$ is a starting point for counting problems, such as the asymptotic number of cylinders of closed geodesics on a typical surface.\nThe passage from ergodicity of $g_t$ to unique ergodicity of almost every vertical flow uses Masur's criterion.\nIf the orbit $g_t(X, \\omega)$ returns infinitely often to a compact set as $t \\to \\infty$, then the vertical flow is uniquely ergodic.\n",
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