American Institute of Mathematical Sciences

January  2011, 5(1): 71-105. doi: 10.3934/jmd.2011.5.71

Counting closed geodesics in moduli space

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

Received  March 2010 Revised  February 2011 Published  April 2011

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 $R$.
Citation: Alex Eskin, Maryam Mirzakhani. Counting closed geodesics in moduli space. Journal of Modern Dynamics, 2011, 5 (1) : 71-105. doi: 10.3934/jmd.2011.5.71
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