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Logarithm laws for unipotent flows on hyperbolic manifolds
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.
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. |
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. |





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