Isolation, equidistribution, and orbit closures for the $\mathrm{SL}(2,\mathbb{R})$ action on  moduli space

Alex Eskin; Maryam Mirzakhani; Amir Mohammadi

doi:https://doi.org/10.4007/annals.2015.182.2.7

Abstract

We prove results about orbit closures and equidistribution for the $\mathrm{SL}(2,\mathbb{R})$ action on the moduli space of compact Riemann surfaces, which are analogous to the theory of unipotent flows. The proofs of the main theorems rely on the measure classification theorem of the first two authors and a certain isolation property of closed $\mathrm{SL}(2,\mathbb{R})$ invariant manifolds developed in this paper.

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@article{Masur:upper, mrkey = {1053805}, author = {Masur, Howard}, title = {The growth rate of trajectories of a quadratic differential}, journal = {Ergodic Theory Dynam. Systems}, fjournal = {Ergodic Theory and Dynamical Systems}, volume = {10}, year = {1990}, number = {1}, pages = {151--176}, issn = {0143-3857}, mrclass = {30F30 (30F60 58F11 58F17)}, mrnumber = {1053805}, mrreviewer = {Masakazu Shiba}, doi = {10.1017/S0143385700005459}, zblnumber = {0706.30035}, }
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@incollection{Masur:lower, mrkey = {0955824}, author = {Masur, Howard}, title = {Lower bounds for the number of saddle connections and closed trajectories of a quadratic differential}, booktitle = {Holomorphic Functions and Moduli, {V}ol. {I}}, venue = {{B}erkeley, {CA}, 1986}, series = {Math. Sci. Res. Inst. Publ.}, volume = {10}, pages = {215--228}, publisher = {Springer-Verlag}, year = {1988}, mrclass = {30F30}, mrnumber = {0955824}, mrreviewer = {K. Strebel}, doi = {10.1007/978-1-4613-9602-4_20}, address = {New York}, zblnumber = {0661.30034}, }
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Isolation, equidistribution, and orbit closures for the $\mathrm{SL}(2,\mathbb{R})$ action on moduli space

Abstract

Authors