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Counting closed geodesics in strata

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Abstract

We compute the asymptotic growth rate of the number \(N({{\mathcal {C}}}, R)\) of closed geodesics of length \(\le R\) in a connected component \({{\mathcal {C}}}\) of a stratum of quadratic differentials. We prove that, for any \(0\le \theta \le 1\), the number of closed geodesics \(\gamma \) of length at most R such that \(\gamma \) spends at least \(\theta \)-fraction of its time outside of a compact subset of \({{\mathcal {C}}}\) is exponentially smaller than \(N({{\mathcal {C}}}, R)\). The theorem follows from a lattice counting statement. For points x, y in the moduli space \({{{\mathcal {M}}}(S)}\) of Riemann surfaces, and for \(0 \le \theta \le 1\) we find an upper-bound for the number of geodesic paths of length \(\le R\) in \({{\mathcal {C}}}\) which connect a point near x to a point near y and spend at least a \(\theta \)-fraction of the time outside of a compact subset of \({{\mathcal {C}}}\).

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Acknowledgements

We would like to thanks the referee for many useful comments that have improve the exposition of the paper at several places.

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Authors and Affiliations

  1. Department of Mathematics, University of Chicago, Chicago, IL, 60637, USA

    Alex Eskin

  2. Department of Mathematics, Stanford University, Building 380, Stanford, CA, 94305, USA

    Maryam Mirzakhani

  3. Department of Mathematics, University of Toronto, Toronto, ON, M5S 2E4, Canada

    Kasra Rafi

Authors
  1. Alex Eskin
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  2. Maryam Mirzakhani
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  3. Kasra Rafi
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Correspondence to Kasra Rafi.

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Deceased: Maryam Mirzakhani.

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Eskin, A., Mirzakhani, M. & Rafi, K. Counting closed geodesics in strata. Invent. math. 215, 535–607 (2019). https://doi.org/10.1007/s00222-018-0832-y

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