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Counting closed geodesics in strata

  • Alex Eskin
  • Maryam Mirzakhani
  • Kasra Rafi
Article
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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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Notes

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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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA
  2. 2.Department of MathematicsStanford UniversityStanfordUSA
  3. 3.Department of MathematicsUniversity of TorontoTorontoCanada

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