# 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), C. R. Acad. Sci. Paris Sér. I Math., 29 (1981), 75-78. [2] A. Avila, S. Gouezel and J.-C. Yoccoz, Exponential mixing for the Teichmüller flow, Publ. Math. IHES, 104 (2006), 143-211. doi: 10.1007/s10240-006-0001-5. [3] A. Avila and M. Resende, Exponential mixing for the Teichmüller flow in the space of quadratic differentials,, Preprint, (). [4] J. Athreya, Quantitative recurrence and large deviations for Teichmüeller geodesic flow, Geom. Dedicata, 119 (2006), 121-140. doi: 10.1007/s10711-006-9058-z. [5] J. Athreya, A. Bufetov, A. Eskin and M. Mirzakhani, Lattice point asymptotics and volume growth on Teichmüller space,, Preprint, (). [6] L. Bers, An extremal problem for quasiconformal maps and a theorem by Thurston, Acta Math., 141 (1978), 73-98. doi: 10.1007/BF02545743. [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), 245-261, back matter. [8] P. Buser, "Geometry and Spectra of Compact Riemann Surfaces," Progr. Math., 106, Birkhäuser, Boston, Inc., Boston, MA, 1992. [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), 93-141. doi: 10.2307/120984. [10] A. Eskin and H. Masur, Asymptotic formulas on flat surfaces, Ergodic Theory Dynam. Systems, 21 (2001), 443-478. doi: 10.1017/S0143385701001225. [11] B. Farb and D. Margalit, A primer on mapping-class groups,, \url{http://www.math.utah.edu/ margalit/primer}., (). [12] A. Fathi, F. Laudenbach and V. Poénaru, Travaux de Thurston sur les surfaces, Asterisque, Vols. 66 and 67 (1979). [13] G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2), 155 (2002), 1-103. doi: 10.2307/3062150. [14] U. Hamenstädt, Bernoulli measures for the Teichmüller flow,, Preprint, (). [15] U. Hamenstädt, Bowen's construction for the Teichmüller flow,, Preprint, (). [16] U. Hamenstädt, Dynamics of the Teichmüller flow on compact invariant sets, J. Mod. Dynamics, 4 (2010), 393-418. doi: 10.3934/jmd.2010.4.393. [17] J. L. Harer and R. C. Penner, "Combinatorics of Train Tracks," Annals of Mathematics Studies, 125, Princeton University Press, Princeton, NJ, 1992. [18] J. Hubbard, "Teichmüller Theory and Applications to Geometry, Topology, and Dynamics," Vol. 1, Teichmüller theory. With contributions by Adrien Douady, William Dunbar, Roland Roeder, Sylvain Bonnot, David Brown, Allen Hatcher, Chris Hruska and Sudeb Mitra. With forewords by William Thurston and Clifford Earle. Matrix Editions, Ithaca, NY, 2006. [19] N. V. Ivanov, Coefficients of expansion of pseudo-Anosov homeomorphisms, (Russian) Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 167 (1988), Issled. Topol. 6, 111-116, 191; translation in J. Soviet Math., 52 (1990), 2819-2822 doi: 10.1007/BF01099245. [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, 54, Cambridge University Press, Cambridge, 1995. [21] S. Kerckhoff, The asymptotic geometry of Teichmüller space, Topology, 19 (1980), 23-41. doi: 10.1016/0040-9383(80)90029-4. [22] G. A. Margulis, "On Some Aspects of the Theory of Anosov Systems," With a survey by Richard Sharp: Periodic orbits of hyperbolic flows, Translated from the Russian by Valentina Vladimirovna Szulikowska, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004. [23] B. Maskit, Comparison of hyperbolic and extremal lengths, Ann. Acad. Sci. Fenn. Ser. A I Math., 10 (1985), 381-386. [24] H. Masur, Interval-exchange transformations and measured foliations, Ann. of Math. (2), 115 (1982), 169-200. [25] H. Masur and J. Smillie, Hausdorff Dimension of sets of nonergodic measured foliations, Ann. of Math. (2), 134 (1991), 455-543. doi: 10.2307/2944356. [26] Y. Minsky, Extremal length estimates and product regions in Teichmüller space, Duke Math. J., 83 (1996), 249-286. doi: 10.1215/S0012-7094-96-08310-6. [27] K. Rafi, Closed geodesics in the thin part of moduli space,, In preparation., (). [28] K. Rafi, Thick-thin decomposition of quadratic differentials, Math. Res. Lett., 14 (2007), 333-341. [29] W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.), 19 (1988), 417-431. [30] W. Veech, The Teichmüller geodesic flow, Ann. of Math. (2), 124 (1986), 441-530. doi: 10.2307/2007091.

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##### References:
 [1] P. Arnoux and J. Yoccoz, Construction de difféomorphismes pseudo-Anosov, (French), C. R. Acad. Sci. Paris Sér. I Math., 29 (1981), 75-78. [2] A. Avila, S. Gouezel and J.-C. Yoccoz, Exponential mixing for the Teichmüller flow, Publ. Math. IHES, 104 (2006), 143-211. doi: 10.1007/s10240-006-0001-5. [3] A. Avila and M. Resende, Exponential mixing for the Teichmüller flow in the space of quadratic differentials,, Preprint, (). [4] J. Athreya, Quantitative recurrence and large deviations for Teichmüeller geodesic flow, Geom. Dedicata, 119 (2006), 121-140. doi: 10.1007/s10711-006-9058-z. [5] J. Athreya, A. Bufetov, A. Eskin and M. Mirzakhani, Lattice point asymptotics and volume growth on Teichmüller space,, Preprint, (). [6] L. Bers, An extremal problem for quasiconformal maps and a theorem by Thurston, Acta Math., 141 (1978), 73-98. doi: 10.1007/BF02545743. [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), 245-261, back matter. [8] P. Buser, "Geometry and Spectra of Compact Riemann Surfaces," Progr. Math., 106, Birkhäuser, Boston, Inc., Boston, MA, 1992. [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), 93-141. doi: 10.2307/120984. [10] A. Eskin and H. Masur, Asymptotic formulas on flat surfaces, Ergodic Theory Dynam. Systems, 21 (2001), 443-478. doi: 10.1017/S0143385701001225. [11] B. Farb and D. Margalit, A primer on mapping-class groups,, \url{http://www.math.utah.edu/ margalit/primer}., (). [12] A. Fathi, F. Laudenbach and V. Poénaru, Travaux de Thurston sur les surfaces, Asterisque, Vols. 66 and 67 (1979). [13] G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2), 155 (2002), 1-103. doi: 10.2307/3062150. [14] U. Hamenstädt, Bernoulli measures for the Teichmüller flow,, Preprint, (). [15] U. Hamenstädt, Bowen's construction for the Teichmüller flow,, Preprint, (). [16] U. Hamenstädt, Dynamics of the Teichmüller flow on compact invariant sets, J. Mod. Dynamics, 4 (2010), 393-418. doi: 10.3934/jmd.2010.4.393. [17] J. L. Harer and R. C. Penner, "Combinatorics of Train Tracks," Annals of Mathematics Studies, 125, Princeton University Press, Princeton, NJ, 1992. [18] J. Hubbard, "Teichmüller Theory and Applications to Geometry, Topology, and Dynamics," Vol. 1, Teichmüller theory. With contributions by Adrien Douady, William Dunbar, Roland Roeder, Sylvain Bonnot, David Brown, Allen Hatcher, Chris Hruska and Sudeb Mitra. With forewords by William Thurston and Clifford Earle. Matrix Editions, Ithaca, NY, 2006. [19] N. V. Ivanov, Coefficients of expansion of pseudo-Anosov homeomorphisms, (Russian) Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 167 (1988), Issled. Topol. 6, 111-116, 191; translation in J. Soviet Math., 52 (1990), 2819-2822 doi: 10.1007/BF01099245. [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, 54, Cambridge University Press, Cambridge, 1995. [21] S. Kerckhoff, The asymptotic geometry of Teichmüller space, Topology, 19 (1980), 23-41. doi: 10.1016/0040-9383(80)90029-4. [22] G. A. Margulis, "On Some Aspects of the Theory of Anosov Systems," With a survey by Richard Sharp: Periodic orbits of hyperbolic flows, Translated from the Russian by Valentina Vladimirovna Szulikowska, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004. [23] B. Maskit, Comparison of hyperbolic and extremal lengths, Ann. Acad. Sci. Fenn. Ser. A I Math., 10 (1985), 381-386. [24] H. Masur, Interval-exchange transformations and measured foliations, Ann. of Math. (2), 115 (1982), 169-200. [25] H. Masur and J. Smillie, Hausdorff Dimension of sets of nonergodic measured foliations, Ann. of Math. (2), 134 (1991), 455-543. doi: 10.2307/2944356. [26] Y. Minsky, Extremal length estimates and product regions in Teichmüller space, Duke Math. J., 83 (1996), 249-286. doi: 10.1215/S0012-7094-96-08310-6. [27] K. Rafi, Closed geodesics in the thin part of moduli space,, In preparation., (). [28] K. Rafi, Thick-thin decomposition of quadratic differentials, Math. Res. Lett., 14 (2007), 333-341. [29] W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.), 19 (1988), 417-431. [30] W. Veech, The Teichmüller geodesic flow, Ann. of Math. (2), 124 (1986), 441-530. doi: 10.2307/2007091.
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