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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 |
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, (). Google Scholar |
[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, (). Google Scholar |
[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}., (). Google Scholar |
[12] |
A. Fathi, F. Laudenbach and V. Poénaru, Travaux de Thurston sur les surfaces,, Asterisque, 66 (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.
doi: 10.2307/3062150. |
[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.
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., (). Google Scholar |
[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, (). Google Scholar |
[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, (). Google Scholar |
[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}., (). Google Scholar |
[12] |
A. Fathi, F. Laudenbach and V. Poénaru, Travaux de Thurston sur les surfaces,, Asterisque, 66 (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.
doi: 10.2307/3062150. |
[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.
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., (). Google Scholar |
[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. |
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