Masur–Veech theorem

Remarks

The stratum $\mathcal{H}(k_1, \dots, k_n)$ of all such surfaces, of any area, has local period coordinates. Take 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. Then the map $(X, \omega) \mapsto (\int_{\gamma_j} \omega)_j \in \C^{2g + n - 1}$ is a local chart. The 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 $1$. The 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 $0 < r \le 1$. The 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.

Masur 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. Consequence: for $\nu$-almost every translation surface, the vertical straight-line flow is uniquely ergodic. Through the first return map to a transversal, this gives Keane's conjecture: almost every interval exchange transformation with an irreducible permutation is uniquely ergodic.

Going deeper

For a student

The strata $\mathcal{H}_1(k_1, \dots, k_n)$ are moduli spaces of abelian differentials, fibred over the moduli space of Riemann surfaces of genus $g$. The flow $g_t$ projects to geodesics for the Teichmüller metric on Teichmüller space. Ergodicity of $g_t$ is a starting point for counting problems, such as the asymptotic number of cylinders of closed geodesics on a typical surface. The passage from ergodicity of $g_t$ to unique ergodicity of almost every vertical flow uses Masur's criterion. If the orbit $g_t(X, \omega)$ returns infinitely often to a compact set as $t \to \infty$, then the vertical flow is uniquely ergodic.