Journal of Modern Dynamics (JMD)

Counting closed geodesics in moduli space

Pages: 71 - 105, Issue 1, January 2011      doi:10.3934/jmd.2011.5.71

       Abstract        References        Full Text (396.8K)              Related Articles

Alex Eskin - Department of Mathematics, University of Chicago, Chicago, IL 60637, United States (email)
Maryam Mirzakhani - Department of Mathematics, Stanford University, Stanford, CA 94305, United States (email)

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 $R$.

Keywords:  Closed geodesics, Teichm├╝ller space, Moduli space.
Mathematics Subject Classification:  Primary: 37A25; Secondary: 30F60.

Received: March 2010;      Revised: February 2011;      Available Online: April 2011.