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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J. Athreya, "Quantitative recurrence and large deviations for Teichmuller geodesic flow," Geom. Dedicata, vol. 119, pp. 121-140, 2006.
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author = {Athreya, Jayadev},
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[ABEM]
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@article{ABEM, mrkey = {2913101},
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mrclass = {32G15 (30F60 37D40)},
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doi = {10.1215/00127094-1548443},
zblnumber = {1246.37009},
} -
[Avila:Eskin:Moeller:yeti]
A. Avila, A. Eskin, and M. Moeller, Symplectic and Isometric SL$(2,{R})$ invariant subbundles of the Hodge bundle.
@misc{Avila:Eskin:Moeller:yeti, key={AEM15},
author = {Avila, A. and Eskin, A. and Moeller, M.},
title = {Symplectic and Isometric {SL}$(2,{R})$ invariant subbundles of the {H}odge bundle},
sortyear = {2015},
doi= {10.1515/crelle-2014-0142},
note = {{\em J. Reine Angew. Math.},
published online April 14, 2015}
} -
[Athreya:Eskin:Zorich] J. Athreya, A. Eskin, and A. Zorich, Rectangular billiards and volumes of spaces of quadratic differentials on $\mathbb{C}P^1$ (with an appendix by Jon Chaika), 2012.
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title = {Rectangular billiards and volumes of spaces of quadratic differentials on {$\mathbb{C}P^1$} (with an appendix by {J}on {C}haika)},
arxiv = {1212.1660},
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} -
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[Eskin:Marklof:Morris]
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[EMas]
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