Counting saddle connections in a homology class modulo <inline-formula><tex-math id="M1">\begin{document} \boldsymbol q \end{document}</tex-math></inline-formula> (with an appendix by Rodolfo Gutiérrez-Romo)

Previous Article
Topological proof of Benoist-Quint's orbit closure theorem for $ \boldsymbol{ \operatorname{SO}(d, 1)} $
JMD Home
This Volume
Next Article
The local-global principle for integral Soddy sphere packings

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.
[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, 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.
[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.
[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.
[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.
[7]	M. Burger, Kazhdan constants for SL(3, Z), J. Reine Angew. Math., 413 (1991), 36-67. doi: 10.1515/crll.1991.413.36.
[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.
[9]	A. Eskin and H. Masur, Asymptotic formulas on flat surfaces, Ergodic Theory Dynam. Systems, 21 (2001), 443-478. doi: 10.1017/S0143385701001225.
[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.
[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.
[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.
[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.
[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.
[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.
[16]	A. Lubotzky, Discrete Groups, Expanding Graphs and Invariant Measures, Springer Science and Business Media, 2010. doi: 10.1007/978-3-0346-0332-4.
[17]	M. Magee, On Selberg's Eigenvalue Conjecture for moduli spaces of abelian differentials, arXiv: 1609.05500, 2018.
[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.
[19]	H. Masur, Interval exchange transformations and measured foliations, Ann. of Math. (2), 115 (1982), 169-200. doi: 10.2307/1971341.
[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.
[21]	H. Masur, The growth rate of trajectories of a quadratic differential, Ergodic Theory Dynam. Systems, 10 (1990), 151-176. doi: 10.1017/S0143385700005459.
[22]	A. Nevo, R. Rühr and B. Weiss, Effective counting on translation surfaces, arXiv: 1708.06263, 2017.
[23]	A. Rapinchuk, Strong approximation for algebraic groups, Thin Groups and Superstrong Approximation, 61 (2014), 269-298.
[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.
[25]	W. A. Veech, The Teichmüller geodesic flow, Ann. of Math. (2), 124 (1986), 441-530. doi: 10.2307/2007091.
[26]	W. A. Veech, Siegel measures, Ann. of Math. (2), 148 (1998), 895-944. doi: 10.2307/121033.
[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.
[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.

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.
[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, 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.
[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.
[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.
[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.
[7]	M. Burger, Kazhdan constants for SL(3, Z), J. Reine Angew. Math., 413 (1991), 36-67. doi: 10.1515/crll.1991.413.36.
[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.
[9]	A. Eskin and H. Masur, Asymptotic formulas on flat surfaces, Ergodic Theory Dynam. Systems, 21 (2001), 443-478. doi: 10.1017/S0143385701001225.
[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.
[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.
[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.
[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.
[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.
[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.
[16]	A. Lubotzky, Discrete Groups, Expanding Graphs and Invariant Measures, Springer Science and Business Media, 2010. doi: 10.1007/978-3-0346-0332-4.
[17]	M. Magee, On Selberg's Eigenvalue Conjecture for moduli spaces of abelian differentials, arXiv: 1609.05500, 2018.
[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.
[19]	H. Masur, Interval exchange transformations and measured foliations, Ann. of Math. (2), 115 (1982), 169-200. doi: 10.2307/1971341.
[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.
[21]	H. Masur, The growth rate of trajectories of a quadratic differential, Ergodic Theory Dynam. Systems, 10 (1990), 151-176. doi: 10.1017/S0143385700005459.
[22]	A. Nevo, R. Rühr and B. Weiss, Effective counting on translation surfaces, arXiv: 1708.06263, 2017.
[23]	A. Rapinchuk, Strong approximation for algebraic groups, Thin Groups and Superstrong Approximation, 61 (2014), 269-298.
[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.
[25]	W. A. Veech, The Teichmüller geodesic flow, Ann. of Math. (2), 124 (1986), 441-530. doi: 10.2307/2007091.
[26]	W. A. Veech, Siegel measures, Ann. of Math. (2), 148 (1998), 895-944. doi: 10.2307/121033.
[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.
[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.

[1]	C. Alonso-González, M. I. Camacho, F. Cano. Topological classification of multiple saddle connections. Discrete and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 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 and 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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 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 and 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 and Continuous Dynamical Systems, 2020, 40 (7) : 4533-4564. doi: 10.3934/dcds.2020190
[12]	Luís Simão Ferreira. A lower bound for the spectral gap of the conjugate Kac process with 3 interacting particles. Kinetic and Related Models, 2022, 15 (1) : 91-117. doi: 10.3934/krm.2021045
[13]	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 and Optimization, 2017, 7 (1) : 77-88. doi: 10.3934/naco.2017004
[14]	Nikolai Dokuchaev. On strong causal binomial approximation for stochastic processes. Discrete and Continuous Dynamical Systems - B, 2014, 19 (6) : 1549-1562. doi: 10.3934/dcdsb.2014.19.1549
[15]	Eduardo Lara, Rodolfo Rodríguez, Pablo Venegas. Spectral approximation of the curl operator in multiply connected domains. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 235-253. doi: 10.3934/dcdss.2016.9.235
[16]	Lam Quoc Anh, Nguyen Van Hung. Gap functions and Hausdorff continuity of solution mappings to parametric strong vector quasiequilibrium problems. Journal of Industrial and Management Optimization, 2018, 14 (1) : 65-79. doi: 10.3934/jimo.2017037
[17]	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 and Applied Analysis, 2020, 19 (5) : 2549-2573. doi: 10.3934/cpaa.2020112
[18]	Stefano Galatolo, Rafael Lucena. Spectral gap and quantitative statistical stability for systems with contracting fibers and Lorenz-like maps. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1309-1360. doi: 10.3934/dcds.2020079
[19]	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 and Management Optimization, 2014, 10 (3) : 977-987. doi: 10.3934/jimo.2014.10.977
[20]	Naoufel Ben Abdallah, Raymond El Hajj. Diffusion and guiding center approximation for particle transport in strong magnetic fields. Kinetic and Related Models, 2008, 1 (3) : 331-354. doi: 10.3934/krm.2008.1.331

2020 Impact Factor: 0.848

Tools

Metrics

Other articles
by authors

on AIMS
Michael Magee Rene Rühr
on Google Scholar
Michael Magee Rene Rühr

2020 Impact Factor: 0.848

Tools

Article outline

2020 Impact Factor: 0.848

Tools

Metrics

Other articles
by authors

on AIMS
Michael Magee Rene Rühr
on Google Scholar
Michael Magee Rene Rühr

[Back to Top]