Counting closed geodesics in moduli space

Alex Eskin; Maryam Mirzakhani

doi:10.3934/jmd.2011.5.71

Article Contents

2011, Volume 5, Issue 1: 71-105. Doi: 10.3934/jmd.2011.5.71 open access

This issue Previous Article Boundary unitary representations-irreducibility and rigidity Next Article Integrability and Lyapunov exponents

Counting closed geodesics in moduli space

Alex Eskin^1, and
Maryam Mirzakhani^2,

1.
Department of Mathematics, University of Chicago, Chicago, IL 60637
2.
Department of Mathematics, Stanford University, Stanford, CA 94305

Received: February 28, 2010

Revised: January 31, 2011

Published: March 2011

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Abstract

We compute the asymptotics, as $R$ tends to infinity, of the number $N(R)$ of closed geodesics of length at most $R$ in the moduli space of compact Riemann surfaces of genus $g$. In fact, $N(R)$ is the number of conjugacy classes of pseudo-Anosov elements of the mapping class group of a compact surface of genus $g$ of translation length at most $R$.

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Mathematics Subject Classification: Primary: 37A25; Secondary: 30F60.

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