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.

[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.

[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.

[4]

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

[5]

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

[6]

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

[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.

[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.

[9]

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

[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.

[11]

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

[12]

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

[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.

[14]

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

[15]

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

[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.

[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.

[18]

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

[19]

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

[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.

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.

[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.

[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.

[4]

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

[5]

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

[6]

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

[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.

[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.

[9]

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

[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.

[11]

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

[12]

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

[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.

[14]

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

[15]

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

[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.

[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.

[18]

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

[19]

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

[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.

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