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. 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. 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. 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. doi: 10.4171/CMH/263.

[5]

R. Bowen, Symbolic dynamics for hyperbolic flows,, Amer. J. Math., 95 (1973), 429. 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. 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), (1987).

[8]

A. Eskin and M. Mirzakhani, Counting closed geodesics in moduli space,, J. Mod. Dyn., 5 (2011), 71. 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, (2006), 187.

[11]

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

[12]

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

[13]

U. Hamenstädt, Stability of quasi-geodesics in Teichmüller space,, Geom. Dedicata, 146 (2010), 101. 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. 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. 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. 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.

[20]

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

[21]

G. Margulis, On Some Aspects of the Theory of Anosov Systems,, Springer Monographs in Mathematics, (2004).

[22]

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

[23]

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

[24]

R. Penner with J. Harer, Combinatorics of Train Tracks,, Ann. Math. Studies, (1992).

[25]

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

[26]

W. Veech, Moduli spaces of quadratic differentials,, J. Analyse Math., 55 (1990), 117. 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. 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. 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. 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. doi: 10.4171/CMH/263.

[5]

R. Bowen, Symbolic dynamics for hyperbolic flows,, Amer. J. Math., 95 (1973), 429. 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. 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), (1987).

[8]

A. Eskin and M. Mirzakhani, Counting closed geodesics in moduli space,, J. Mod. Dyn., 5 (2011), 71. 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, (2006), 187.

[11]

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

[12]

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

[13]

U. Hamenstädt, Stability of quasi-geodesics in Teichmüller space,, Geom. Dedicata, 146 (2010), 101. 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. 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. 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. 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.

[20]

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

[21]

G. Margulis, On Some Aspects of the Theory of Anosov Systems,, Springer Monographs in Mathematics, (2004).

[22]

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

[23]

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

[24]

R. Penner with J. Harer, Combinatorics of Train Tracks,, Ann. Math. Studies, (1992).

[25]

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

[26]

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

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