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

Counting closed geodesics in moduli space

Alex Eskin 1, and  Maryam Mirzakhani 2,

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 $R$.
Keywords: Closed geodesics, Teichmüller space, Moduli space..
Mathematics Subject Classification: Primary: 37A25; Secondary: 30F6.
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.  Google Scholar

[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.  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), 121-140. 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), 73-98. 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), 245-261, back matter.  Google Scholar

[8]

P. Buser, "Geometry and Spectra of Compact Riemann Surfaces," Progr. Math., 106, Birkhäuser, Boston, Inc., Boston, MA, 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), 93-141. doi: 10.2307/120984.  Google Scholar

[10]

A. Eskin and H. Masur, Asymptotic formulas on flat surfaces, Ergodic Theory Dynam. Systems, 21 (2001), 443-478. 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, Vols. 66 and 67 (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), 1-103. 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), 393-418. 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, Princeton University Press, Princeton, NJ, 1992.  Google Scholar

[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.  Google Scholar

[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.  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, 54, Cambridge University Press, Cambridge, 1995.  Google Scholar

[21]

S. Kerckhoff, The asymptotic geometry of Teichmüller space, Topology, 19 (1980), 23-41. 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, Translated from the Russian by Valentina Vladimirovna Szulikowska, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004.  Google Scholar

[23]

B. Maskit, Comparison of hyperbolic and extremal lengths, Ann. Acad. Sci. Fenn. Ser. A I Math., 10 (1985), 381-386.  Google Scholar

[24]

H. Masur, Interval-exchange transformations and measured foliations, Ann. of Math. (2), 115 (1982), 169-200.  Google Scholar

[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.  Google Scholar

[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.  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), 333-341.  Google Scholar

[29]

W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.), 19 (1988), 417-431.  Google Scholar

[30]

W. Veech, The Teichmüller geodesic flow, Ann. of Math. (2), 124 (1986), 441-530. doi: 10.2307/2007091.  Google Scholar

show all references

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.  Google Scholar

[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.  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), 121-140. 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), 73-98. 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), 245-261, back matter.  Google Scholar

[8]

P. Buser, "Geometry and Spectra of Compact Riemann Surfaces," Progr. Math., 106, Birkhäuser, Boston, Inc., Boston, MA, 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), 93-141. doi: 10.2307/120984.  Google Scholar

[10]

A. Eskin and H. Masur, Asymptotic formulas on flat surfaces, Ergodic Theory Dynam. Systems, 21 (2001), 443-478. 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, Vols. 66 and 67 (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), 1-103. 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), 393-418. 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, Princeton University Press, Princeton, NJ, 1992.  Google Scholar

[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.  Google Scholar

[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.  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, 54, Cambridge University Press, Cambridge, 1995.  Google Scholar

[21]

S. Kerckhoff, The asymptotic geometry of Teichmüller space, Topology, 19 (1980), 23-41. 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, Translated from the Russian by Valentina Vladimirovna Szulikowska, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004.  Google Scholar

[23]

B. Maskit, Comparison of hyperbolic and extremal lengths, Ann. Acad. Sci. Fenn. Ser. A I Math., 10 (1985), 381-386.  Google Scholar

[24]

H. Masur, Interval-exchange transformations and measured foliations, Ann. of Math. (2), 115 (1982), 169-200.  Google Scholar

[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.  Google Scholar

[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.  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), 333-341.  Google Scholar

[29]

W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.), 19 (1988), 417-431.  Google Scholar

[30]

W. Veech, The Teichmüller geodesic flow, Ann. of Math. (2), 124 (1986), 441-530. doi: 10.2307/2007091.  Google Scholar

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