2019, 15: 329-343. doi: 10.3934/jmd.2019023

Uniform distribution of saddle connection lengths (with an appendix by Daniel El-Baz and Bingrong Huang)

Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA

Received  August 2018 Revised  May 13, 2019 Published  December 2019

For any $ \mathrm{SL}(2, \mathbb{R}) $ invariant and ergodic probability measure on any stratum of flat surfaces, almost every flat surface has the property that its non-decreasing sequence of saddle connection lengths is uniformly distributed mod one.

Citation: Jon Chaika, Donald Robertson. Uniform distribution of saddle connection lengths (with an appendix by Daniel El-Baz and Bingrong Huang). Journal of Modern Dynamics, 2019, 15: 329-343. doi: 10.3934/jmd.2019023
References:
[1]

J. S. Athreya and J. Chaika, The distribution of gaps for saddle connection directions, Geom. Funct. Anal., 22 (2012), 1491-1516.  doi: 10.1007/s00039-012-0164-9.  Google Scholar

[2]

J. S. Athreya, J. Chaika and S. Lelièvre, The gap distribution of slopes on the golden L, in Recent Trends in Ergodic Theory and Dynamical Systems, Contemp. Math., 631, Amer. Math. Soc., Providence, RI, 2015, 47-62. doi: 10.1090/conm/631/12595.  Google Scholar

[3]

J. S. AthreyaY. Cheung and H. Masur, Siegel-Veech transforms are in L2. With an appendix by Jayadev S. Athreya and Rene Rühr, J. Mod. Dyn., 14 (2019), 1-19.  doi: 10.3934/jmd.2019022.  Google Scholar

[4]

B. Dozier, Convergence of Siegel-Veech constants, Geom. Dedicata, 198 (2019), 131-142.  doi: 10.1007/s10711-018-0332-7.  Google Scholar

[5]

B. Dozier, Equidistribution of saddle connections on translation surfaces, J. Mod. Dyn., 14 (2019), 87-120.  doi: 10.3934/jmd.2019004.  Google Scholar

[6]

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

[7]

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

[8]

A. EskinM. Mirzakhani and A. Mohammadi, Isolation, equidistribution, and orbit closures for the $\mathrm{SL}(2,\mathbb{R})$ action on moduli space, Ann. of Math. (2), 182 (2015), 673-721.  doi: 10.4007/annals.2015.182.2.7.  Google Scholar

[9]

G. Forni, Limits of geodesic push-forwards of horocycle invariant measures, preprint, 2017. Google Scholar

[10]

M. N. Huxley and W. G. Nowak, Primitive lattice points in convex planar domains, Acta Arith., 76 (1996), 271-283.  doi: 10.4064/aa-76-3-271-283.  Google Scholar

[11]

A. W. Knapp, Representation Theory of Semisimple Groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 2001.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

A. NevoR. Rühr and B. Weiss, Effective counting on translation surfaces, Adv. Math., 360 (2020), 106890.  doi: 10.1016/j.aim.2019.106890.  Google Scholar

[16]

C. Uyanik and G. Work, The distribution of gaps for saddle connections on the octagon, Int. Math. Res. Not. IMRN, (2016), 5569-5602.  doi: 10.1093/imrn/rnv317.  Google Scholar

[17]

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

[18]

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

[19]

Y. Vorobets, Periodic geodesics on translation surfaces, preprint, arXiv: math/0307249, 2003. doi: 10.1090/conm/385/07199.  Google Scholar

[20]

Y. Vorobets, Periodic geodesics on generic translation surfaces, in Algebraic and Topological Dynamics, Contemp. Math., 385, Amer. Math. Soc., Providence, RI, 2005,205-258. doi: 10.1090/conm/385/07199.  Google Scholar

show all references

References:
[1]

J. S. Athreya and J. Chaika, The distribution of gaps for saddle connection directions, Geom. Funct. Anal., 22 (2012), 1491-1516.  doi: 10.1007/s00039-012-0164-9.  Google Scholar

[2]

J. S. Athreya, J. Chaika and S. Lelièvre, The gap distribution of slopes on the golden L, in Recent Trends in Ergodic Theory and Dynamical Systems, Contemp. Math., 631, Amer. Math. Soc., Providence, RI, 2015, 47-62. doi: 10.1090/conm/631/12595.  Google Scholar

[3]

J. S. AthreyaY. Cheung and H. Masur, Siegel-Veech transforms are in L2. With an appendix by Jayadev S. Athreya and Rene Rühr, J. Mod. Dyn., 14 (2019), 1-19.  doi: 10.3934/jmd.2019022.  Google Scholar

[4]

B. Dozier, Convergence of Siegel-Veech constants, Geom. Dedicata, 198 (2019), 131-142.  doi: 10.1007/s10711-018-0332-7.  Google Scholar

[5]

B. Dozier, Equidistribution of saddle connections on translation surfaces, J. Mod. Dyn., 14 (2019), 87-120.  doi: 10.3934/jmd.2019004.  Google Scholar

[6]

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

[7]

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

[8]

A. EskinM. Mirzakhani and A. Mohammadi, Isolation, equidistribution, and orbit closures for the $\mathrm{SL}(2,\mathbb{R})$ action on moduli space, Ann. of Math. (2), 182 (2015), 673-721.  doi: 10.4007/annals.2015.182.2.7.  Google Scholar

[9]

G. Forni, Limits of geodesic push-forwards of horocycle invariant measures, preprint, 2017. Google Scholar

[10]

M. N. Huxley and W. G. Nowak, Primitive lattice points in convex planar domains, Acta Arith., 76 (1996), 271-283.  doi: 10.4064/aa-76-3-271-283.  Google Scholar

[11]

A. W. Knapp, Representation Theory of Semisimple Groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 2001.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

A. NevoR. Rühr and B. Weiss, Effective counting on translation surfaces, Adv. Math., 360 (2020), 106890.  doi: 10.1016/j.aim.2019.106890.  Google Scholar

[16]

C. Uyanik and G. Work, The distribution of gaps for saddle connections on the octagon, Int. Math. Res. Not. IMRN, (2016), 5569-5602.  doi: 10.1093/imrn/rnv317.  Google Scholar

[17]

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

[18]

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

[19]

Y. Vorobets, Periodic geodesics on translation surfaces, preprint, arXiv: math/0307249, 2003. doi: 10.1090/conm/385/07199.  Google Scholar

[20]

Y. Vorobets, Periodic geodesics on generic translation surfaces, in Algebraic and Topological Dynamics, Contemp. Math., 385, Amer. Math. Soc., Providence, RI, 2005,205-258. doi: 10.1090/conm/385/07199.  Google Scholar

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