# American Institute of Mathematical Sciences

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. [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. [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. 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. 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. [8] P. Buser, "Geometry and Spectra of Compact Riemann Surfaces,", Progr. Math., (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. doi: 10.2307/120984. [10] A. Eskin and H. Masur, Asymptotic formulas on flat surfaces,, Ergodic Theory Dynam. Systems, 21 (2001), 443. 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, 66 (1979). [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. [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. doi: 10.3934/jmd.2010.4.393. [17] J. L. Harer and R. C. Penner, "Combinatorics of Train Tracks,", Annals of Mathematics Studies, 125 (1992). [18] J. Hubbard, "Teichmüller Theory and Applications to Geometry, Topology, and Dynamics,", Vol. \textbf{1}, 1 (2006). [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. [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). [21] S. Kerckhoff, The asymptotic geometry of Teichmüller space,, Topology, 19 (1980), 23. 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, (2004). [23] B. Maskit, Comparison of hyperbolic and extremal lengths,, Ann. Acad. Sci. Fenn. Ser. A I Math., 10 (1985), 381. [24] H. Masur, Interval-exchange transformations and measured foliations,, Ann. of Math. (2), 115 (1982), 169. [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. [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. [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. [29] W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces,, Bull. Amer. Math. Soc. (N.S.), 19 (1988), 417. [30] W. Veech, The Teichmüller geodesic flow,, Ann. of Math. (2), 124 (1986), 441. doi: 10.2307/2007091.

show all references

##### References:
 [1] P. Arnoux and J. Yoccoz, Construction de difféomorphismes pseudo-Anosov,, (French), 29 (1981), 75. [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. [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. 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. 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. [8] P. Buser, "Geometry and Spectra of Compact Riemann Surfaces,", Progr. Math., (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. doi: 10.2307/120984. [10] A. Eskin and H. Masur, Asymptotic formulas on flat surfaces,, Ergodic Theory Dynam. Systems, 21 (2001), 443. 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, 66 (1979). [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. [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. doi: 10.3934/jmd.2010.4.393. [17] J. L. Harer and R. C. Penner, "Combinatorics of Train Tracks,", Annals of Mathematics Studies, 125 (1992). [18] J. Hubbard, "Teichmüller Theory and Applications to Geometry, Topology, and Dynamics,", Vol. \textbf{1}, 1 (2006). [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. [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). [21] S. Kerckhoff, The asymptotic geometry of Teichmüller space,, Topology, 19 (1980), 23. 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, (2004). [23] B. Maskit, Comparison of hyperbolic and extremal lengths,, Ann. Acad. Sci. Fenn. Ser. A I Math., 10 (1985), 381. [24] H. Masur, Interval-exchange transformations and measured foliations,, Ann. of Math. (2), 115 (1982), 169. [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. [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. [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. [29] W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces,, Bull. Amer. Math. Soc. (N.S.), 19 (1988), 417. [30] W. Veech, The Teichmüller geodesic flow,, Ann. of Math. (2), 124 (1986), 441. doi: 10.2307/2007091.
 [1] Martin Möller. Shimura and Teichmüller curves. Journal of Modern Dynamics, 2011, 5 (1) : 1-32. doi: 10.3934/jmd.2011.5.1 [2] Corentin Boissy. Classification of Rauzy classes in the moduli space of Abelian and quadratic differentials. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3433-3457. doi: 10.3934/dcds.2012.32.3433 [3] Ursula Hamenstädt. Bowen's construction for the Teichmüller flow. Journal of Modern Dynamics, 2013, 7 (4) : 489-526. doi: 10.3934/jmd.2013.7.489 [4] Ursula Hamenstädt. Dynamics of the Teichmüller flow on compact invariant sets. Journal of Modern Dynamics, 2010, 4 (2) : 393-418. doi: 10.3934/jmd.2010.4.393 [5] Fei Yu, Kang Zuo. Weierstrass filtration on Teichmüller curves and Lyapunov exponents. Journal of Modern Dynamics, 2013, 7 (2) : 209-237. doi: 10.3934/jmd.2013.7.209 [6] Dawei Chen. Strata of abelian differentials and the Teichmüller dynamics. Journal of Modern Dynamics, 2013, 7 (1) : 135-152. doi: 10.3934/jmd.2013.7.135 [7] Jonathan Chaika, Yitwah Cheung, Howard Masur. Winning games for bounded geodesics in moduli spaces of quadratic differentials. Journal of Modern Dynamics, 2013, 7 (3) : 395-427. doi: 10.3934/jmd.2013.7.395 [8] Guizhen Cui, Yunping Jiang, Anthony Quas. Scaling functions and Gibbs measures and Teichmüller spaces of circle endomorphisms. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 535-552. doi: 10.3934/dcds.1999.5.535 [9] Chi Po Choi, Xianfeng Gu, Lok Ming Lui. Subdivision connectivity remeshing via Teichmüller extremal map. Inverse Problems & Imaging, 2017, 11 (5) : 825-855. doi: 10.3934/ipi.2017039 [10] Matteo Costantini, André Kappes. The equation of the Kenyon-Smillie (2, 3, 4)-Teichmüller curve. Journal of Modern Dynamics, 2017, 11: 17-41. doi: 10.3934/jmd.2017002 [11] Artur O. Lopes, Rafael O. Ruggiero. Large deviations and Aubry-Mather measures supported in nonhyperbolic closed geodesics. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1155-1174. doi: 10.3934/dcds.2011.29.1155 [12] Wenzhi Luo, Zeév Rudnick, Peter Sarnak. The variance of arithmetic measures associated to closed geodesics on the modular surface. Journal of Modern Dynamics, 2009, 3 (2) : 271-309. doi: 10.3934/jmd.2009.3.271 [13] Giovanni Forni, Carlos Matheus. Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards. Journal of Modern Dynamics, 2014, 8 (3&4) : 271-436. doi: 10.3934/jmd.2014.8.271 [14] Giovanni Forni. On the Brin Prize work of Artur Avila in Teichmüller dynamics and interval-exchange transformations. Journal of Modern Dynamics, 2012, 6 (2) : 139-182. doi: 10.3934/jmd.2012.6.139 [15] Alex Wright. Schwarz triangle mappings and Teichmüller curves: Abelian square-tiled surfaces. Journal of Modern Dynamics, 2012, 6 (3) : 405-426. doi: 10.3934/jmd.2012.6.405 [16] Matthias Rumberger. Lyapunov exponents on the orbit space. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 91-113. doi: 10.3934/dcds.2001.7.91 [17] M. Jotz. The leaf space of a multiplicative foliation. Journal of Geometric Mechanics, 2012, 4 (3) : 313-332. doi: 10.3934/jgm.2012.4.313 [18] Jose-Luis Lisani, Antoni Buades, Jean-Michel Morel. How to explore the patch space. Inverse Problems & Imaging, 2013, 7 (3) : 813-838. doi: 10.3934/ipi.2013.7.813 [19] Feimin Zhong, Jinxing Xie, Jing Jiao. Solutions for bargaining games with incomplete information: General type space and action space. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1-14. doi: 10.3934/jimo.2017084 [20] Ravi Vakil and Aleksey Zinger. A natural smooth compactification of the space of elliptic curves in projective space. Electronic Research Announcements, 2007, 13: 53-59.

2016 Impact Factor: 0.706