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Mixing, counting, and equidistribution in Lie groups
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Asymptotic formulas on flat surfaces
We find asymptotics for the number of cylinders and saddle connections on flat surfaces. These results extend previous results of Veech.
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UPPER BOUNDS AND ASYMPTOTICS IN A QUANTITATIVE VERSION OF THE OPPENHEIM CONJECTURE
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Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow
We compute the sum of the positive Lyapunov exponents of the Hodge bundle with respect to the Teichmuller geodesic flow. The computation is based on the analytic Riemann-Roch Theorem and uses a…
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Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials
We compute the asymptotics of the number of connected branched coverings of a torus as their degree goes to infinity and the ramification type stays fixed. These numbers are equal to the volumes of…
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Quasi-flats and rigidity in higher rank symmetric spaces
In this paper we use elementary geometrical and topological methods to study some questions about the coarse geometry of symmetric spaces. Our results are powerful enough to apply to noncocompact…
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Invariant and stationary measures for the SL(2,R) action on Moduli space
We prove some ergodic-theoretic rigidity properties of the action of SL(2,R) on moduli space. In particular, we show that any ergodic measure invariant under the action of the upper triangular…
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Moduli spaces of Abelian differentials: The principal boundary, counting problems, and the Siegel–Veech constants
A holomorphic 1-form on a compact Riemann surface S naturally defines a flat metric on S with cone-type singularities. We present the following surprising phenomenon: having found a geodesic segment…
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Counting closed geodesics in moduli space
We compute the asymptotics, as R tends to infinity, of the number of closed geodesics in Moduli space of length at most R, or equivalently the number of pseudo-Anosov elements of the mapping class…
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Invariant and stationary measures for the action on Moduli space
We prove some ergodic-theoretic rigidity properties of the action of on moduli space. In particular, we show that any ergodic measure invariant under the action of the upper triangular subgroup of is…
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