American Institute of Mathematical Sciences

Advanced
  2019, 15: 237-262. doi: 10.3934/jmd.2019020

Counting saddle connections in a homology class modulo $ \boldsymbol q $ (with an appendix by Rodolfo Gutiérrez-Romo)

Michael Magee 1, and  Rene Rühr 2,

1. 

Department of Mathematical Sciences, Durham University, Lower Mountjoy, Stockton Rd, Durham DH1 3LE, United Kingdom
2. 

Faculty of Mathematics, Technion, Haifa, 32000 Israel

(with an appendix by Rodolfo Gutiérrez-Romo)
Rodolfo Gutiérrez-Romo < rodolfo.gutierrez@imj-prg.fr>: Institut de Mathématiques de Jussieu - Paris Rive Gauche, UMR 7586, Bátiment Sophie Germain, 75205 PARIS Cedex 13, France

Received  November 30, 2018 Revised  May 10, 2019 Published  August 2019

Fund Project: Supported in part by the S.N.F., project number 168823.

We give effective estimates for the number of saddle connections on a translation surface that have length $ \leq L $ and are in a prescribed homology class modulo $ q $. Our estimates apply to almost all translation surfaces in a stratum of the moduli space of translation surfaces, with respect to the Masur–Veech measure on the stratum.

Keywords: Saddle connections, effective counting, congruences, strong approximation, spectral gap.
Mathematics Subject Classification: Primary: 32G15, 11N45; Secondary: 37E35, 20H05, 37A30.
Citation: Michael Magee, Rene Rühr. Counting saddle connections in a homology class modulo $ \boldsymbol q $ (with an appendix by Rodolfo Gutiérrez-Romo). Journal of Modern Dynamics, 2019, 15: 237-262. doi: 10.3934/jmd.2019020
References:
[1]

J. S. Athreya, Y. Cheung and H. Masur, Siegel–Veech transforms are in $L^{2}$, J. Mod. Dyn., 14 (2019), 1-19.   Google Scholar

[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.  Google Scholar

[3]

A. Avila, C. Matheus and J.-C. Yoccoz, Zorich conjecture for hyperelliptic Rauzy-Veech groups, Math. Ann., 370 (2018), 785-809.  doi: 10.1007/s00208-017-1568-5.  Google Scholar

[4]

A. Avila and M. Viana, Simplicity of Lyapunov spectra: Proof of the Zorich-Kontsevich conjecture, Acta Math., 198 (2007), 1-56.  doi: 10.1007/s11511-007-0012-1.  Google Scholar

[5]

B. Bekka, P. de la Harpe and A. Valette, Kazhdan's Property (T), New Mathematical Monographs, 11, Cambridge University Press, Cambridge, 2008. doi: 10.1017/CBO9780511542749.  Google Scholar

[6]

E. Breuillard and H. Oh, editors, Thin Groups and Superstrong Approximation, Mathematical Sciences Research Institute Publications, 61, Cambridge University Press, Cambridge, 2014; Selected expanded papers from the workshop held in Berkeley, CA, February 6–10, 2012.  Google Scholar

[7]

M. Burger, Kazhdan constants for SL(3, Z), J. Reine Angew. Math., 413 (1991), 36-67.  doi: 10.1515/crll.1991.413.36.  Google Scholar

[8]

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

[9]

A. Eskin and H. Masur, Asymptotic formulas on flat surfaces, Ergodic Theory Dynam. Systems, 21 (2001), 443-478.  doi: 10.1017/S0143385701001225.  Google Scholar

[10]

A. Eskin, H. Masur and A. Zorich, Moduli spaces of abelian differentials: The principal boundary, counting problems, and the Siegel-Veech constants, Publ. Math. Inst. Hautes Études Sci., 97 (2003), 61-179.  doi: 10.1007/s10240-003-0015-1.  Google Scholar

[11]

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.  Google Scholar

[12]

R. Gutiérrez-Romo, Classification of Rauzy–Veech groups: Proof of the Zorich conjecture, Invent. Math., 215 (2019), 741-778.  doi: 10.1007/s00222-018-0836-7.  Google Scholar

[13]

D. A. Každan, On the connection of the dual space of a group with the structure of its closed subgroups, Funkcional. Anal. i Priložen., 1 (1967), 71-74.   Google Scholar

[14]

B. Kirkwood and B. R. McDonald, The symplectic group over a ring with one in its stable range, Pacific J. Math., 92 (1981), 111-125.  doi: 10.2140/pjm.1981.92.111.  Google Scholar

[15]

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.  Google Scholar

[16]

A. Lubotzky, Discrete Groups, Expanding Graphs and Invariant Measures, Springer Science and Business Media, 2010. doi: 10.1007/978-3-0346-0332-4.  Google Scholar

[17]

M. Magee, On Selberg's Eigenvalue Conjecture for moduli spaces of abelian differentials, arXiv: 1609.05500, 2018. Google Scholar

[18]

G. A. Margulis, Indefinite quadratic forms and unipotent flows on homogeneous spaces, in Dynamical Systems and Ergodic Theory (Warsaw, 1986), Banach Center Publ., 23, PWN, Warsaw, 1989,399–409.  Google Scholar

[19]

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

[20]

H. Masur, Lower bounds for the number of saddle connections and closed trajectories of a quadratic differential, in Holomorphic Functions and Moduli, Vol. I (Berkeley, CA, 1986), Math. Sci. Res. Inst. Publ., 10, Springer, New York, 1988,215–228. doi: 10.1007/978-1-4613-9602-4_20.  Google Scholar

[21]

H. Masur, The growth rate of trajectories of a quadratic differential, Ergodic Theory Dynam. Systems, 10 (1990), 151-176.  doi: 10.1017/S0143385700005459.  Google Scholar

[22]

A. Nevo, R. Rühr and B. Weiss, Effective counting on translation surfaces, arXiv: 1708.06263, 2017. Google Scholar

[23]

A. Rapinchuk, Strong approximation for algebraic groups, Thin Groups and Superstrong Approximation, 61 (2014), 269-298.   Google Scholar

[24]

W. A. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2), 115 (1982), 201-242.  doi: 10.2307/1971391.  Google Scholar

[25]

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

[26]

W. A. Veech, Siegel measures, Ann. of Math. (2), 148 (1998), 895-944.  doi: 10.2307/121033.  Google Scholar

[27]

J.-C. Yoccoz, Interval exchange maps and translation surfaces, in Homogeneous Flows, Moduli Spaces and Arithmetic, Clay Math. Proc., 10, Amer. Math. Soc., Providence, RI, 2010, 1–69.  Google Scholar

[28]

A. Zorich, How do the leaves of a closed 1-form wind around a surface?, in Pseudoperiodic Topology, Amer. Math. Soc. Transl. Ser. 2,197, Amer. Math. Soc., Providence, RI, 1999,135–178. doi: 10.1090/trans2/197/05.  Google Scholar

show all references

References:
[1]

J. S. Athreya, Y. Cheung and H. Masur, Siegel–Veech transforms are in $L^{2}$, J. Mod. Dyn., 14 (2019), 1-19.   Google Scholar

[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.  Google Scholar

[3]

A. Avila, C. Matheus and J.-C. Yoccoz, Zorich conjecture for hyperelliptic Rauzy-Veech groups, Math. Ann., 370 (2018), 785-809.  doi: 10.1007/s00208-017-1568-5.  Google Scholar

[4]

A. Avila and M. Viana, Simplicity of Lyapunov spectra: Proof of the Zorich-Kontsevich conjecture, Acta Math., 198 (2007), 1-56.  doi: 10.1007/s11511-007-0012-1.  Google Scholar

[5]

B. Bekka, P. de la Harpe and A. Valette, Kazhdan's Property (T), New Mathematical Monographs, 11, Cambridge University Press, Cambridge, 2008. doi: 10.1017/CBO9780511542749.  Google Scholar

[6]

E. Breuillard and H. Oh, editors, Thin Groups and Superstrong Approximation, Mathematical Sciences Research Institute Publications, 61, Cambridge University Press, Cambridge, 2014; Selected expanded papers from the workshop held in Berkeley, CA, February 6–10, 2012.  Google Scholar

[7]

M. Burger, Kazhdan constants for SL(3, Z), J. Reine Angew. Math., 413 (1991), 36-67.  doi: 10.1515/crll.1991.413.36.  Google Scholar

[8]

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

[9]

A. Eskin and H. Masur, Asymptotic formulas on flat surfaces, Ergodic Theory Dynam. Systems, 21 (2001), 443-478.  doi: 10.1017/S0143385701001225.  Google Scholar

[10]

A. Eskin, H. Masur and A. Zorich, Moduli spaces of abelian differentials: The principal boundary, counting problems, and the Siegel-Veech constants, Publ. Math. Inst. Hautes Études Sci., 97 (2003), 61-179.  doi: 10.1007/s10240-003-0015-1.  Google Scholar

[11]

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.  Google Scholar

[12]

R. Gutiérrez-Romo, Classification of Rauzy–Veech groups: Proof of the Zorich conjecture, Invent. Math., 215 (2019), 741-778.  doi: 10.1007/s00222-018-0836-7.  Google Scholar

[13]

D. A. Každan, On the connection of the dual space of a group with the structure of its closed subgroups, Funkcional. Anal. i Priložen., 1 (1967), 71-74.   Google Scholar

[14]

B. Kirkwood and B. R. McDonald, The symplectic group over a ring with one in its stable range, Pacific J. Math., 92 (1981), 111-125.  doi: 10.2140/pjm.1981.92.111.  Google Scholar

[15]

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.  Google Scholar

[16]

A. Lubotzky, Discrete Groups, Expanding Graphs and Invariant Measures, Springer Science and Business Media, 2010. doi: 10.1007/978-3-0346-0332-4.  Google Scholar

[17]

M. Magee, On Selberg's Eigenvalue Conjecture for moduli spaces of abelian differentials, arXiv: 1609.05500, 2018. Google Scholar

[18]

G. A. Margulis, Indefinite quadratic forms and unipotent flows on homogeneous spaces, in Dynamical Systems and Ergodic Theory (Warsaw, 1986), Banach Center Publ., 23, PWN, Warsaw, 1989,399–409.  Google Scholar

[19]

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

[20]

H. Masur, Lower bounds for the number of saddle connections and closed trajectories of a quadratic differential, in Holomorphic Functions and Moduli, Vol. I (Berkeley, CA, 1986), Math. Sci. Res. Inst. Publ., 10, Springer, New York, 1988,215–228. doi: 10.1007/978-1-4613-9602-4_20.  Google Scholar

[21]

H. Masur, The growth rate of trajectories of a quadratic differential, Ergodic Theory Dynam. Systems, 10 (1990), 151-176.  doi: 10.1017/S0143385700005459.  Google Scholar

[22]

A. Nevo, R. Rühr and B. Weiss, Effective counting on translation surfaces, arXiv: 1708.06263, 2017. Google Scholar

[23]

A. Rapinchuk, Strong approximation for algebraic groups, Thin Groups and Superstrong Approximation, 61 (2014), 269-298.   Google Scholar

[24]

W. A. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2), 115 (1982), 201-242.  doi: 10.2307/1971391.  Google Scholar

[25]

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

[26]

W. A. Veech, Siegel measures, Ann. of Math. (2), 148 (1998), 895-944.  doi: 10.2307/121033.  Google Scholar

[27]

J.-C. Yoccoz, Interval exchange maps and translation surfaces, in Homogeneous Flows, Moduli Spaces and Arithmetic, Clay Math. Proc., 10, Amer. Math. Soc., Providence, RI, 2010, 1–69.  Google Scholar

[28]

A. Zorich, How do the leaves of a closed 1-form wind around a surface?, in Pseudoperiodic Topology, Amer. Math. Soc. Transl. Ser. 2,197, Amer. Math. Soc., Providence, RI, 1999,135–178. doi: 10.1090/trans2/197/05.  Google Scholar

[1]

C. Alonso-González, M. I. Camacho, F. Cano. Topological classification of multiple saddle connections. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 395-414. doi: 10.3934/dcds.2006.15.395
[2]

Benjamin Dozier. Equidistribution of saddle connections on translation surfaces. Journal of Modern Dynamics, 2019, 14: 87-120. doi: 10.3934/jmd.2019004
[3]

Alexandre A. P. Rodrigues. Moduli for heteroclinic connections involving saddle-foci and periodic solutions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3155-3182. doi: 10.3934/dcds.2015.35.3155
[4]

Bruno Colbois, Alexandre Girouard. The spectral gap of graphs and Steklov eigenvalues on surfaces. Electronic Research Announcements, 2014, 21: 19-27. doi: 10.3934/era.2014.21.19
[5]

Damien Thomine. A spectral gap for transfer operators of piecewise expanding maps. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 917-944. doi: 10.3934/dcds.2011.30.917
[6]

Rúben Sousa, Semyon Yakubovich. The spectral expansion approach to index transforms and connections with the theory of diffusion processes. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2351-2378. doi: 10.3934/cpaa.2018112
[7]

Francisco Arana-Herrera. Counting square-tiled surfaces with prescribed real and imaginary foliations and connections to Mirzakhani's asymptotics for simple closed hyperbolic geodesics. Journal of Modern Dynamics, 2020, 16: 81-107. doi: 10.3934/jmd.2020004
[8]

Sébastien Gouëzel. An interval map with a spectral gap on Lipschitz functions, but not on bounded variation functions. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1205-1208. doi: 10.3934/dcds.2009.24.1205
[9]

Jean-Pierre Conze, Y. Guivarc'h. Ergodicity of group actions and spectral gap, applications to random walks and Markov shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4239-4269. doi: 10.3934/dcds.2013.33.4239
[10]

Soña Pavlíková, Daniel Ševčovič. On construction of upper and lower bounds for the HOMO-LUMO spectral gap. Numerical Algebra, Control & Optimization, 2019, 9 (1) : 53-69. doi: 10.3934/naco.2019005
[11]

Shuang Chen, Jun Shen. Large spectral gap induced by small delay and its application to reduction. Discrete & Continuous Dynamical Systems - A, 2020, 40 (7) : 4533-4564. doi: 10.3934/dcds.2020190
[12]

Z. K. Eshkuvatov, M. Kammuji, Bachok M. Taib, N. M. A. Nik Long. Effective approximation method for solving linear Fredholm-Volterra integral equations. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 77-88. doi: 10.3934/naco.2017004
[13]

Eduardo Lara, Rodolfo Rodríguez, Pablo Venegas. Spectral approximation of the curl operator in multiply connected domains. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 235-253. doi: 10.3934/dcdss.2016.9.235
[14]

Nikolai Dokuchaev. On strong causal binomial approximation for stochastic processes. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1549-1562. doi: 10.3934/dcdsb.2014.19.1549
[15]

Andrea Bondesan, Laurent Boudin, Marc Briant, Bérénice Grec. Stability of the spectral gap for the Boltzmann multi-species operator linearized around non-equilibrium maxwell distributions. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2549-2573. doi: 10.3934/cpaa.2020112
[16]

Stefano Galatolo, Rafael Lucena. Spectral gap and quantitative statistical stability for systems with contracting fibers and Lorenz-like maps. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1309-1360. doi: 10.3934/dcds.2020079
[17]

Lam Quoc Anh, Nguyen Van Hung. Gap functions and Hausdorff continuity of solution mappings to parametric strong vector quasiequilibrium problems. Journal of Industrial & Management Optimization, 2018, 14 (1) : 65-79. doi: 10.3934/jimo.2017037
[18]

Suxiang He, Pan Zhang, Xiao Hu, Rong Hu. A sample average approximation method based on a D-gap function for stochastic variational inequality problems. Journal of Industrial & Management Optimization, 2014, 10 (3) : 977-987. doi: 10.3934/jimo.2014.10.977
[19]

Naoufel Ben Abdallah, Raymond El Hajj. Diffusion and guiding center approximation for particle transport in strong magnetic fields. Kinetic & Related Models, 2008, 1 (3) : 331-354. doi: 10.3934/krm.2008.1.331
[20]

Hao Yang, Fuke Wu, Peter E. Kloeden. Existence and approximation of strong solutions of SDEs with fractional diffusion coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5553-5567. doi: 10.3934/dcdsb.2019071

2019 Impact Factor: 0.465

Tools

Metrics

  • PDF downloads (72)
  • HTML views (384)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences