# 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
##### References:
 [1] P. Arnoux and J. Yoccoz, Construction de difféomorphismes pseudo-Anosov,, (French), 29 (1981), 75.   Google Scholar [2] A. Avila, S. Gouezel and J.-C. Yoccoz, Exponential mixing for the Teichmüller flow,, Publ. Math. IHES, 104 (2006), 143.  doi: 10.1007/s10240-006-0001-5.  Google Scholar [3] A. Avila and M. Resende, Exponential mixing for the Teichmüller flow in the space of quadratic differentials,, Preprint, ().   Google Scholar [4] J. Athreya, Quantitative recurrence and large deviations for Teichmüeller geodesic flow,, Geom. Dedicata, 119 (2006), 121.  doi: 10.1007/s10711-006-9058-z.  Google Scholar [5] J. Athreya, A. Bufetov, A. Eskin and M. Mirzakhani, Lattice point asymptotics and volume growth on Teichmüller space,, Preprint, ().   Google Scholar [6] L. Bers, An extremal problem for quasiconformal maps and a theorem by Thurston,, Acta Math., 141 (1978), 73.  doi: 10.1007/BF02545743.  Google Scholar [7] A. Bufetov, Logarithmic asymptotics for the number of periodic orbits of the Teichmueller flow on Veech's space of zippered rectangles,, Mosc. Math. J., 9 (2009), 245.   Google Scholar [8] P. Buser, "Geometry and Spectra of Compact Riemann Surfaces,", Progr. Math., (1992).   Google Scholar [9] A. Eskin, G. Margulis and S. Mozes, Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture,, Ann. of Math. (2), 147 (1998), 93.  doi: 10.2307/120984.  Google Scholar [10] A. Eskin and H. Masur, Asymptotic formulas on flat surfaces,, Ergodic Theory Dynam. Systems, 21 (2001), 443.  doi: 10.1017/S0143385701001225.  Google Scholar [11] B. Farb and D. Margalit, A primer on mapping-class groups,, \url{http://www.math.utah.edu/ margalit/primer}., ().   Google Scholar [12] A. Fathi, F. Laudenbach and V. Poénaru, Travaux de Thurston sur les surfaces,, Asterisque, 66 (1979).   Google Scholar [13] G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus,, Ann. of Math. (2), 155 (2002), 1.  doi: 10.2307/3062150.  Google Scholar [14] U. Hamenstädt, Bernoulli measures for the Teichmüller flow,, Preprint, ().   Google Scholar [15] U. Hamenstädt, Bowen's construction for the Teichmüller flow,, Preprint, ().   Google Scholar [16] U. Hamenstädt, Dynamics of the Teichmüller flow on compact invariant sets,, J. Mod. Dynamics, 4 (2010), 393.  doi: 10.3934/jmd.2010.4.393.  Google Scholar [17] J. L. Harer and R. C. Penner, "Combinatorics of Train Tracks,", Annals of Mathematics Studies, 125 (1992).   Google Scholar [18] J. Hubbard, "Teichmüller Theory and Applications to Geometry, Topology, and Dynamics,", Vol. \textbf{1}, 1 (2006).   Google Scholar [19] N. V. Ivanov, Coefficients of expansion of pseudo-Anosov homeomorphisms,, (Russian) Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) \textbf{167} (1988), 167 (1988), 111.  doi: 10.1007/BF01099245.  Google Scholar [20] A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,", With a supplementary chapter by Katok and Leonardo Mendoza. Encyclopedia of Mathematics and its Applications, (1995).   Google Scholar [21] S. Kerckhoff, The asymptotic geometry of Teichmüller space,, Topology, 19 (1980), 23.  doi: 10.1016/0040-9383(80)90029-4.  Google Scholar [22] G. A. Margulis, "On Some Aspects of the Theory of Anosov Systems,", With a survey by Richard Sharp: Periodic orbits of hyperbolic flows, (2004).   Google Scholar [23] B. Maskit, Comparison of hyperbolic and extremal lengths,, Ann. Acad. Sci. Fenn. Ser. A I Math., 10 (1985), 381.   Google Scholar [24] H. Masur, Interval-exchange transformations and measured foliations,, Ann. of Math. (2), 115 (1982), 169.   Google Scholar [25] H. Masur and J. Smillie, Hausdorff Dimension of sets of nonergodic measured foliations,, Ann. of Math. (2), 134 (1991), 455.  doi: 10.2307/2944356.  Google Scholar [26] Y. Minsky, Extremal length estimates and product regions in Teichmüller space,, Duke Math. J., 83 (1996), 249.  doi: 10.1215/S0012-7094-96-08310-6.  Google Scholar [27] K. Rafi, Closed geodesics in the thin part of moduli space,, In preparation., ().   Google Scholar [28] K. Rafi, Thick-thin decomposition of quadratic differentials,, Math. Res. Lett., 14 (2007), 333.   Google Scholar [29] W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces,, Bull. Amer. Math. Soc. (N.S.), 19 (1988), 417.   Google Scholar [30] W. Veech, The Teichmüller geodesic flow,, Ann. of Math. (2), 124 (1986), 441.  doi: 10.2307/2007091.  Google Scholar

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##### References:
 [1] P. Arnoux and J. Yoccoz, Construction de difféomorphismes pseudo-Anosov,, (French), 29 (1981), 75.   Google Scholar [2] A. Avila, S. Gouezel and J.-C. Yoccoz, Exponential mixing for the Teichmüller flow,, Publ. Math. IHES, 104 (2006), 143.  doi: 10.1007/s10240-006-0001-5.  Google Scholar [3] A. Avila and M. Resende, Exponential mixing for the Teichmüller flow in the space of quadratic differentials,, Preprint, ().   Google Scholar [4] J. Athreya, Quantitative recurrence and large deviations for Teichmüeller geodesic flow,, Geom. Dedicata, 119 (2006), 121.  doi: 10.1007/s10711-006-9058-z.  Google Scholar [5] J. Athreya, A. Bufetov, A. Eskin and M. Mirzakhani, Lattice point asymptotics and volume growth on Teichmüller space,, Preprint, ().   Google Scholar [6] L. Bers, An extremal problem for quasiconformal maps and a theorem by Thurston,, Acta Math., 141 (1978), 73.  doi: 10.1007/BF02545743.  Google Scholar [7] A. Bufetov, Logarithmic asymptotics for the number of periodic orbits of the Teichmueller flow on Veech's space of zippered rectangles,, Mosc. Math. J., 9 (2009), 245.   Google Scholar [8] P. Buser, "Geometry and Spectra of Compact Riemann Surfaces,", Progr. Math., (1992).   Google Scholar [9] A. Eskin, G. Margulis and S. Mozes, Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture,, Ann. of Math. (2), 147 (1998), 93.  doi: 10.2307/120984.  Google Scholar [10] A. Eskin and H. Masur, Asymptotic formulas on flat surfaces,, Ergodic Theory Dynam. Systems, 21 (2001), 443.  doi: 10.1017/S0143385701001225.  Google Scholar [11] B. Farb and D. Margalit, A primer on mapping-class groups,, \url{http://www.math.utah.edu/ margalit/primer}., ().   Google Scholar [12] A. Fathi, F. Laudenbach and V. Poénaru, Travaux de Thurston sur les surfaces,, Asterisque, 66 (1979).   Google Scholar [13] G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus,, Ann. of Math. (2), 155 (2002), 1.  doi: 10.2307/3062150.  Google Scholar [14] U. Hamenstädt, Bernoulli measures for the Teichmüller flow,, Preprint, ().   Google Scholar [15] U. Hamenstädt, Bowen's construction for the Teichmüller flow,, Preprint, ().   Google Scholar [16] U. Hamenstädt, Dynamics of the Teichmüller flow on compact invariant sets,, J. Mod. Dynamics, 4 (2010), 393.  doi: 10.3934/jmd.2010.4.393.  Google Scholar [17] J. L. Harer and R. C. Penner, "Combinatorics of Train Tracks,", Annals of Mathematics Studies, 125 (1992).   Google Scholar [18] J. Hubbard, "Teichmüller Theory and Applications to Geometry, Topology, and Dynamics,", Vol. \textbf{1}, 1 (2006).   Google Scholar [19] N. V. Ivanov, Coefficients of expansion of pseudo-Anosov homeomorphisms,, (Russian) Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) \textbf{167} (1988), 167 (1988), 111.  doi: 10.1007/BF01099245.  Google Scholar [20] A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,", With a supplementary chapter by Katok and Leonardo Mendoza. Encyclopedia of Mathematics and its Applications, (1995).   Google Scholar [21] S. Kerckhoff, The asymptotic geometry of Teichmüller space,, Topology, 19 (1980), 23.  doi: 10.1016/0040-9383(80)90029-4.  Google Scholar [22] G. A. Margulis, "On Some Aspects of the Theory of Anosov Systems,", With a survey by Richard Sharp: Periodic orbits of hyperbolic flows, (2004).   Google Scholar [23] B. Maskit, Comparison of hyperbolic and extremal lengths,, Ann. Acad. Sci. Fenn. Ser. A I Math., 10 (1985), 381.   Google Scholar [24] H. Masur, Interval-exchange transformations and measured foliations,, Ann. of Math. (2), 115 (1982), 169.   Google Scholar [25] H. Masur and J. Smillie, Hausdorff Dimension of sets of nonergodic measured foliations,, Ann. of Math. (2), 134 (1991), 455.  doi: 10.2307/2944356.  Google Scholar [26] Y. Minsky, Extremal length estimates and product regions in Teichmüller space,, Duke Math. J., 83 (1996), 249.  doi: 10.1215/S0012-7094-96-08310-6.  Google Scholar [27] K. Rafi, Closed geodesics in the thin part of moduli space,, In preparation., ().   Google Scholar [28] K. Rafi, Thick-thin decomposition of quadratic differentials,, Math. Res. Lett., 14 (2007), 333.   Google Scholar [29] W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces,, Bull. Amer. Math. Soc. (N.S.), 19 (1988), 417.   Google Scholar [30] W. Veech, The Teichmüller geodesic flow,, Ann. of Math. (2), 124 (1986), 441.  doi: 10.2307/2007091.  Google Scholar
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