October  2013, 7(4): 489-526. doi: 10.3934/jmd.2013.7.489

Bowen's construction for the Teichmüller flow

1. 

Mathematisches Institut der Rheinischen Friedrich-Wilhelms-Universität Bonn, Endenicher Allee 60, D-53115 Bonn

Received  August 2011 Published  March 2013

Let ${\cal Q}$ be a connected component of a stratum in the moduli space of abelian or quadratic differentials for a nonexceptional Riemann surface $S$ of finite type. We prove that the probability measure on ${\cal Q}$ in the Lebesgue measure class which is invariant under the Teichmüller flow is obtained by Bowen's construction.
Citation: 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
References:
[1]

J. Athreya, A. Bufetov, A. Eskin and M. Mirzakhani, Lattice point asymptotics and volume growth on Teichmüller space, Duke Math. J., 161 (2012), 1055-1111. doi: 10.1215/00127094-1548443.

[2]

A. Avila, S. Gouëzel and J.-C. Yoccoz, Exponential mixing for the Teichmüller flow, Publ. Math. Inst. Hautes Études Sci., 104 (2006), 143-211. doi: 10.1007/s10240-006-0001-5.

[3]

A. Avila and S. Gouëzel, Small eigenvalues of the Laplacian for algebraic measures in moduli space, and mixing properties of the Teichmüller flow, Ann. Math. (2), 178 (2013), 385-442. doi: 10.4007/annals.2013.178.2.1.

[4]

A. Avila and M. J. Resende, Exponential mixing for the Teichmüller flow in the space of quadratic differentials, Comm. Math. Helv., 87 (2012), 589-638. doi: 10.4171/CMH/263.

[5]

R. Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math., 95 (1973), 429-460. doi: 10.2307/2373793.

[6]

A. Bufetov and B. Gurevich, Existence and uniqueness of the measure of maximal entropy for the Teichmüller flow on the moduli space of Abelian differentials, Sb. Math., 202 (2011), 935-970. doi: 10.1070/SM2011v202n07ABEH004172.

[7]

R. Canary, D. Epstein and P. Green, Notes on notes of Thurston, in Analytical and Geometric Aspects of Hyperbolic Space (ed. D. Epstein), London Math. Soc. Lecture Note Ser., 111, Cambridge University Press, Cambridge, 1987.

[8]

A. Eskin and M. Mirzakhani, Counting closed geodesics in moduli space, J. Mod. Dyn., 5 (2011), 71-105. doi: 10.3934/jmd.2011.5.71.

[9]

A. Eskin, M. Mirzakhani and K. Rafi, Counting closed geodesics in strata,, , (). 

[10]

U. Hamenstädt, Train tracks and the Gromov boundary of the complex of curves, in Spaces of Kleinian Groups (eds. Y. Minsky, M. Sakuma and C. Series), London Math. Soc. Lec. Note Ser., 329, Cambridge Univ. Press, Cambridge, (2006), 187-207.

[11]

U. Hamenstädt, Geometry of the mapping class groups. I. Boundary amenability, Invent. Math., 175 (2009), 545-609. doi: 10.1007/s00222-008-0158-2.

[12]

U. Hamenstädt, Invariant Radon measures on measured lamination space, Invent. Math., 176 (2009), 223-273. doi: 10.1007/s00222-008-0163-5.

[13]

U. Hamenstädt, Stability of quasi-geodesics in Teichmüller space, Geom. Dedicata, 146 (2010), 101-116. doi: 10.1007/s10711-009-9428-4.

[14]

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.

[15]

U. Hamenstädt, Symbolic dynamics for the Teichmüller flow,, , (). 

[16]

J. Hubbard and H. Masur, Quadratic differentials and foliations, Acta Math., 142 (1979), 221-274. doi: 10.1007/BF02395062.

[17]

E. Klarreich, The boundary at infinity of the curve complex and the relative Teichmüller space, unpublished manuscript, 1999.

[18]

M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., 153 (2003), 631-678. doi: 10.1007/s00222-003-0303-x.

[19]

E. Lanneau, Connected components of the strata of the moduli spaces of quadratic differentials, Ann. Sci. Éc. Norm. Supér. (4), 41 (2008), 1-56.

[20]

G. Levitt, Foliations and laminations on hyperbolic surfaces, Topology, 22 (1983), 119-135. doi: 10.1016/0040-9383(83)90023-X.

[21]

G. Margulis, On Some Aspects of the Theory of Anosov Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004.

[22]

H. Masur, Interval exchange transformations and measured foliations, Ann. Math. (2), 115 (1982), 169-200. doi: 10.2307/1971341.

[23]

H. Masur and Y. Minsky, Geometry of the complex of curves. I. Hyperbolicity, Invent. Math., 138 (1999), 103-149. doi: 10.1007/s002220050343.

[24]

R. Penner with J. Harer, Combinatorics of Train Tracks, Ann. Math. Studies, 125, Princeton University Press, Princeton, NJ, 1992.

[25]

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

[26]

W. Veech, Moduli spaces of quadratic differentials, J. Analyse Math., 55 (1990), 117-171. doi: 10.1007/BF02789200.

show all references

References:
[1]

J. Athreya, A. Bufetov, A. Eskin and M. Mirzakhani, Lattice point asymptotics and volume growth on Teichmüller space, Duke Math. J., 161 (2012), 1055-1111. doi: 10.1215/00127094-1548443.

[2]

A. Avila, S. Gouëzel and J.-C. Yoccoz, Exponential mixing for the Teichmüller flow, Publ. Math. Inst. Hautes Études Sci., 104 (2006), 143-211. doi: 10.1007/s10240-006-0001-5.

[3]

A. Avila and S. Gouëzel, Small eigenvalues of the Laplacian for algebraic measures in moduli space, and mixing properties of the Teichmüller flow, Ann. Math. (2), 178 (2013), 385-442. doi: 10.4007/annals.2013.178.2.1.

[4]

A. Avila and M. J. Resende, Exponential mixing for the Teichmüller flow in the space of quadratic differentials, Comm. Math. Helv., 87 (2012), 589-638. doi: 10.4171/CMH/263.

[5]

R. Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math., 95 (1973), 429-460. doi: 10.2307/2373793.

[6]

A. Bufetov and B. Gurevich, Existence and uniqueness of the measure of maximal entropy for the Teichmüller flow on the moduli space of Abelian differentials, Sb. Math., 202 (2011), 935-970. doi: 10.1070/SM2011v202n07ABEH004172.

[7]

R. Canary, D. Epstein and P. Green, Notes on notes of Thurston, in Analytical and Geometric Aspects of Hyperbolic Space (ed. D. Epstein), London Math. Soc. Lecture Note Ser., 111, Cambridge University Press, Cambridge, 1987.

[8]

A. Eskin and M. Mirzakhani, Counting closed geodesics in moduli space, J. Mod. Dyn., 5 (2011), 71-105. doi: 10.3934/jmd.2011.5.71.

[9]

A. Eskin, M. Mirzakhani and K. Rafi, Counting closed geodesics in strata,, , (). 

[10]

U. Hamenstädt, Train tracks and the Gromov boundary of the complex of curves, in Spaces of Kleinian Groups (eds. Y. Minsky, M. Sakuma and C. Series), London Math. Soc. Lec. Note Ser., 329, Cambridge Univ. Press, Cambridge, (2006), 187-207.

[11]

U. Hamenstädt, Geometry of the mapping class groups. I. Boundary amenability, Invent. Math., 175 (2009), 545-609. doi: 10.1007/s00222-008-0158-2.

[12]

U. Hamenstädt, Invariant Radon measures on measured lamination space, Invent. Math., 176 (2009), 223-273. doi: 10.1007/s00222-008-0163-5.

[13]

U. Hamenstädt, Stability of quasi-geodesics in Teichmüller space, Geom. Dedicata, 146 (2010), 101-116. doi: 10.1007/s10711-009-9428-4.

[14]

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.

[15]

U. Hamenstädt, Symbolic dynamics for the Teichmüller flow,, , (). 

[16]

J. Hubbard and H. Masur, Quadratic differentials and foliations, Acta Math., 142 (1979), 221-274. doi: 10.1007/BF02395062.

[17]

E. Klarreich, The boundary at infinity of the curve complex and the relative Teichmüller space, unpublished manuscript, 1999.

[18]

M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., 153 (2003), 631-678. doi: 10.1007/s00222-003-0303-x.

[19]

E. Lanneau, Connected components of the strata of the moduli spaces of quadratic differentials, Ann. Sci. Éc. Norm. Supér. (4), 41 (2008), 1-56.

[20]

G. Levitt, Foliations and laminations on hyperbolic surfaces, Topology, 22 (1983), 119-135. doi: 10.1016/0040-9383(83)90023-X.

[21]

G. Margulis, On Some Aspects of the Theory of Anosov Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004.

[22]

H. Masur, Interval exchange transformations and measured foliations, Ann. Math. (2), 115 (1982), 169-200. doi: 10.2307/1971341.

[23]

H. Masur and Y. Minsky, Geometry of the complex of curves. I. Hyperbolicity, Invent. Math., 138 (1999), 103-149. doi: 10.1007/s002220050343.

[24]

R. Penner with J. Harer, Combinatorics of Train Tracks, Ann. Math. Studies, 125, Princeton University Press, Princeton, NJ, 1992.

[25]

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

[26]

W. Veech, Moduli spaces of quadratic differentials, J. Analyse Math., 55 (1990), 117-171. doi: 10.1007/BF02789200.

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