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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 |
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.
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
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References:
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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. |
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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.
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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. |
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J. Bourgain, P. Sarnak and Z. Rudnick, Local statistics of lattice points on the sphere, arXiv: 1204.0134, 2012. |
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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. |
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K. J. Falconer, The Geometry of Fractal Sets, Cambridge Tracts in Mathematics, 85, Cambridge University Press, Cambridge, 1986. |
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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. |
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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. 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. |
| [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. |
| [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 2.
The family of $C_{\gamma}$ 's (hued circles)
Figure 5.
Reflecting a circle
Figure 6.
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Figure 7.
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