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

The gap distribution of directions in some Schottky groups

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.

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.

[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.

[3]

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

[4]

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

[5]

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

[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.

[7]

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

[8]

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

[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.

[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.

[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.

[12]

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

[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.

[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.

[15]

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

[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.

[17]

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

[18]

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

[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.

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 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.

[3]

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

[4]

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

[5]

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

[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.

[7]

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

[8]

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

[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.

[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.

[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.

[12]

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

[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.

[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.

[15]

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

[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.

[17]

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

[18]

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

[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.

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