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  2017, 11: 477-499. doi: 10.3934/jmd.2017019

The gap distribution of directions in some Schottky groups

Xin Zhang

Department of Mathematics, University of Illinois, 1409 W, Green Street, Urbana, IL 61801, USA

Received  April 22, 2016 Revised  May 24, 2017 Published  November 2017

We prove the existence and some properties of the limiting gap distribution for the directions of some Schottky group orbits in the Poincaré disk. A key feature is that the fundamental domains for these groups have infinite area.

Keywords: Fuchsian groups of the second kind, Patterson-Sullivan measure, gap distribution.
Mathematics Subject Classification: Primary: 11J71; Secondary: 11N45, 37F35, 22E40.
Citation: Xin Zhang. The gap distribution of directions in some Schottky groups. Journal of Modern Dynamics, 2017, 11: 477-499. doi: 10.3934/jmd.2017019
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 and Y. Cheung, A Poincaré section for the horocycle flow on the space of lattices, Int. Math. Res. Not. IMRN, 10 (2014), 2643-2690.  doi: 10.1093/imrn/rnt003.  Google Scholar

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F. P. Boca, A. A. Popa and A. Zaharescu, Pair correlation of hyperbolic lattice angles, Int. J. Number Theory, 10 (2014), 1955-1989.  doi: 10.1142/S1793042114500651.  Google Scholar

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J. Bourgain, A. Kontorovich and P. Sarnak, Sector estimates for hyperbolic isometries, Geom. Funct. Anal., 20 (2010), 1175-1200.  doi: 10.1007/s00039-010-0092-5.  Google Scholar

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J. Bourgain, P. Sarnak and Z. Rudnick, Local statistics of lattice points on the sphere, arXiv: 1204.0134, 2012. Google Scholar

[6]

N. D. Elkies and C. T. McMullen, Gaps in ${\sqrt n}\bmod 1$ and ergodic theory, Duke Math. J., 123 (2004), 95-139.  doi: 10.1215/S0012-7094-04-12314-0.  Google Scholar

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K. J. Falconer, The Geometry of Fractal Sets, Cambridge Tracts in Mathematics, 85, Cambridge University Press, Cambridge, 1986.  Google Scholar

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A. Good, Local Analysis of Selberg’s Trace Formula, Lecture Notes in Mathematics, 1040, Springer-Verlag, Berlin, 1983. doi: 10.1007/BFb0073074.  Google Scholar

[9]

D. Kelmer and A. Kontorovich, On the pair correlation density for hyperbolic angles, Duke Math. J., 164 (2015), 473-509.  doi: 10.1215/00127094-2861495.  Google Scholar

[10]

J. Marklof, The $n$-point correlations between values of a linear form, Ergodic Theory Dynam. Systems, 20 (2000), 1127-1172.  doi: 10.1017/S0143385700000626.  Google Scholar

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J. Marklof and A. Str, The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems, Ann. of Math. (2), 172 (2010), 1949-2033.  doi: 10.4007/annals.2010.172.1949.  Google Scholar

[12]

J. Marklof and I. Vinogradov, Directions in Hyperbolic Lattices, arXiv: 1409.3764, 2015. doi: 10.1515/crelle-2015-0070.  Google Scholar

[13]

C. T. McMullen, Hausdorff dimension and conformal dynamics. Ⅲ. Computation of dimension, Amer. J. Math., 120 (1998), 691-721.  doi: 10.1353/ajm.1998.0031.  Google Scholar

[14]

H. Oh and N. A. Shah, Equidistribution and counting for orbits of geometrically finite hyperbolic groups, J. Amer. Math. Soc., 26 (2013), 511-562.  doi: 10.1090/S0894-0347-2012-00749-8.  Google Scholar

[15]

S. J. Patterson, The limit set of a Fuchsian group, Acta Math., 136 (1976), 241-273.  doi: 10.1007/BF02392046.  Google Scholar

[16]

M. S. Risager and A. Södergren, Angles in hyperbolic lattices: The pair correlation density, Trans. Amer. Math. Soc., 369 (2017), 2807-2841.  doi: 10.1090/tran/6770.  Google Scholar

[17]

Z. Rudnick and X. Zhang, Gap distributions in circle packings, Münster J. Math., 10 (2017), 131-170.   Google Scholar

[18]

D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 171-202.   Google Scholar

[19]

D. Sullivan, Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups, Acta Math., 153 (1984), 259-277.  doi: 10.1007/BF02392379.  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 and Y. Cheung, A Poincaré section for the horocycle flow on the space of lattices, Int. Math. Res. Not. IMRN, 10 (2014), 2643-2690.  doi: 10.1093/imrn/rnt003.  Google Scholar

[3]

F. P. Boca, A. A. Popa and A. Zaharescu, Pair correlation of hyperbolic lattice angles, Int. J. Number Theory, 10 (2014), 1955-1989.  doi: 10.1142/S1793042114500651.  Google Scholar

[4]

J. Bourgain, A. Kontorovich and P. Sarnak, Sector estimates for hyperbolic isometries, Geom. Funct. Anal., 20 (2010), 1175-1200.  doi: 10.1007/s00039-010-0092-5.  Google Scholar

[5]

J. Bourgain, P. Sarnak and Z. Rudnick, Local statistics of lattice points on the sphere, arXiv: 1204.0134, 2012. Google Scholar

[6]

N. D. Elkies and C. T. McMullen, Gaps in ${\sqrt n}\bmod 1$ and ergodic theory, Duke Math. J., 123 (2004), 95-139.  doi: 10.1215/S0012-7094-04-12314-0.  Google Scholar

[7]

K. J. Falconer, The Geometry of Fractal Sets, Cambridge Tracts in Mathematics, 85, Cambridge University Press, Cambridge, 1986.  Google Scholar

[8]

A. Good, Local Analysis of Selberg’s Trace Formula, Lecture Notes in Mathematics, 1040, Springer-Verlag, Berlin, 1983. doi: 10.1007/BFb0073074.  Google Scholar

[9]

D. Kelmer and A. Kontorovich, On the pair correlation density for hyperbolic angles, Duke Math. J., 164 (2015), 473-509.  doi: 10.1215/00127094-2861495.  Google Scholar

[10]

J. Marklof, The $n$-point correlations between values of a linear form, Ergodic Theory Dynam. Systems, 20 (2000), 1127-1172.  doi: 10.1017/S0143385700000626.  Google Scholar

[11]

J. Marklof and A. Str, The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems, Ann. of Math. (2), 172 (2010), 1949-2033.  doi: 10.4007/annals.2010.172.1949.  Google Scholar

[12]

J. Marklof and I. Vinogradov, Directions in Hyperbolic Lattices, arXiv: 1409.3764, 2015. doi: 10.1515/crelle-2015-0070.  Google Scholar

[13]

C. T. McMullen, Hausdorff dimension and conformal dynamics. Ⅲ. Computation of dimension, Amer. J. Math., 120 (1998), 691-721.  doi: 10.1353/ajm.1998.0031.  Google Scholar

[14]

H. Oh and N. A. Shah, Equidistribution and counting for orbits of geometrically finite hyperbolic groups, J. Amer. Math. Soc., 26 (2013), 511-562.  doi: 10.1090/S0894-0347-2012-00749-8.  Google Scholar

[15]

S. J. Patterson, The limit set of a Fuchsian group, Acta Math., 136 (1976), 241-273.  doi: 10.1007/BF02392046.  Google Scholar

[16]

M. S. Risager and A. Södergren, Angles in hyperbolic lattices: The pair correlation density, Trans. Amer. Math. Soc., 369 (2017), 2807-2841.  doi: 10.1090/tran/6770.  Google Scholar

[17]

Z. Rudnick and X. Zhang, Gap distributions in circle packings, Münster J. Math., 10 (2017), 131-170.   Google Scholar

[18]

D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 171-202.   Google Scholar

[19]

D. Sullivan, Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups, Acta Math., 153 (1984), 259-277.  doi: 10.1007/BF02392379.  Google Scholar

Figure 1.  A fundamental domain of a group of hyperbolic reflections
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Figure 2.  The family of $C_{\gamma}$ 's (hued circles)
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Figure 3.  The plot for the gap distribution function $F_{10^4,\partial{\mathbb{D}}}$ , for the example illustrated in Figure 2
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Figure 4.  The histograms of $F_{T,\mathscr{I}}'$ of different $T$ , for the example illustrated in Figure 2
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Figure 5.  Reflecting a circle
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Figure 6.  Case (a)
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Figure 7.  Case (b)
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