# American Institue of Mathematical Sciences

2017, 11: 189-217. doi: 10.3934/jmd.2017009

## Effective equidistribution of circles in the limit sets of Kleinian groups

 Mathematics Department, Yale University, New Haven, CT 06520, USA

Received  November 25, 2015 Revised  October 22, 2016 Published  February 2017

Consider a general circle packing $\mathscr{P}$ in the complex plane $\mathbb{C}$ invariant under a Kleinian group $\Gamma$. When $\Gamma$ is convex cocompact or its critical exponent is greater than 1, we obtain an effective equidistribution for small circles in $\mathscr{P}$ intersecting any bounded connected regular set in $\mathbb{C}$; this provides an effective version of an earlier work of Oh-Shah [12]. In view of the recent result of McMullen-Mohammadi-Oh [6], our effective circle counting theorem applies to the circles contained in the limit set of a convex cocompact but non-cocompact Kleinian group whose limit set contains at least one circle. Moreover, consider the circle packing $\mathscr{P}(\mathscr{T})$ of the ideal triangle attained by filling in largest inner circles. We give an effective estimate to the number of disks whose hyperbolic areas are greater than $t$, as $t\to0$, effectivizing the work of Oh [10].

Citation: Wenyu Pan. Effective equidistribution of circles in the limit sets of Kleinian groups. Journal of Modern Dynamics, 2017, 11: 189-217. doi: 10.3934/jmd.2017009
##### References:
 [1] C. J. Bishop, P. W. Jones, Hausdorff dimension and Kleinian groups, Acta Math., 179 (1997), 1-39. doi: 10.1007/BF02392718. [2] R. D. Canary, E. Taylor, Kleinian groups with small limit sets, Duke Math J., 73 (1994), 371-381. doi: 10.1215/S0012-7094-94-07316-X. [3] A. Kontorovich, H. Oh, Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds. With an appendix by Oh and Nimish Shah, J. Amer. Math. Soc., 24 (2011), 603-648. doi: 10.1090/S0894-0347-2011-00691-7. [4] M. Lee, H. Oh, Effective count for Apollonian circle packings and closed horospheres, Geom. Funct. Anal., 23 (2013), 580-621. doi: 10.1007/s00039-013-0217-8. [5] C. McMullen, Hausdorff dimension and conformal dynamics. Ⅲ. Computation of dimension, Amer. J. Math., 120 (1998), 691-721. doi: 10.1353/ajm.1998.0031. [6] C. McMullen, A. Mohammadi and H. Oh, Geodesic planes in hyperbolic 3-manifolds, H. Invent. Math. , (2016). Available from: http://gauss.math.yale.edu/~ho2/. [7] A. Mohammadi, H. Oh, Matrix coefficients, counting and primes for orbits of geometrically finite groups, J. Eur. Math. Soc.(JEMS), 17 (2015), 837-897. doi: 10.4171/JEMS/520. [8] A. Mohammadi, H. Oh, Ergodicity of unipotent flows and Kleinian groups, J. Amer. Math. Soc., 28 (2015), 531-577. doi: 10.1090/S0894-0347-2014-00811-0. [9] D. Mumford, C. Series, D. Wright, Indra's Pearls. The Vision of Felix Klein, Cambridge University Press, New York, 2002. doi: 10.1017/CBO9781107050051.024. [10] H. Oh, Harmonic analysis, ergodic theory and counting for thin groups, in Thin Groups and Superstrong Approximation (eds. E. Breulliard and H. Oh), Math. Sci. Res. Inst. Publ. , 61, Cambridge Univ. Press. , Cambridge, 2014. [11] H. Oh, N. Shah, Equidistribution and counting for orbits of geometrically finite hyper bolic groups, J. Amer. Math. Soc., 26 (2013), 511-562. doi: 10.1090/S0894-0347-2012-00749-8. [12] H. Oh, N. Shah, The asymptotic distribution of circles in the orbits of Kleinian groups, Invent. Math., 187 (2012), 1-35. doi: 10.1007/s00222-011-0326-7. [13] J. Parkkonen and F. Paulin, Counting common perpendicular arcs in negative curvature, To appear in Ergodic Theory and Dynamical Systems, arXiv: 1305.1332. [14] P. Vesselin, L. Stoyanov, Distribution of periods of closed trajectories in exponentially shrinking intervals, Comm. Math. Phys., 310 (2012), 675-704. doi: 10.1007/s00220-012-1419-x. [15] B. Stratmann, Urbański M., Diophantine extremality of the Patterson measure, Math. Proc. Cambridge Philos. Soc., 140 (2006), 297-304. doi: 10.1017/S0305004105009114. [16] B. Stratmann, S. L. Velani, The Patterson measure for geometrically finite groups with parabolic elements. new and old, Proc. London Math. Soc. (3), 71 (1995), 197-220. doi: 10.1112/plms/s3-71.1.197. [17] 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. [18] L. Stoyanov, Spectra of Ruelle transfer operators for axiom A flows, Nonlinearity, 24 (2011), 1089-1120. doi: 10.1088/0951-7715/24/4/005. [19] I. Vinogradov, Effective bisector estimate with applications to Apollonian circle packings, Int. Math. Res. Not. IMRN 2014,3217–3262.

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##### References:
 [1] C. J. Bishop, P. W. Jones, Hausdorff dimension and Kleinian groups, Acta Math., 179 (1997), 1-39. doi: 10.1007/BF02392718. [2] R. D. Canary, E. Taylor, Kleinian groups with small limit sets, Duke Math J., 73 (1994), 371-381. doi: 10.1215/S0012-7094-94-07316-X. [3] A. Kontorovich, H. Oh, Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds. With an appendix by Oh and Nimish Shah, J. Amer. Math. Soc., 24 (2011), 603-648. doi: 10.1090/S0894-0347-2011-00691-7. [4] M. Lee, H. Oh, Effective count for Apollonian circle packings and closed horospheres, Geom. Funct. Anal., 23 (2013), 580-621. doi: 10.1007/s00039-013-0217-8. [5] C. McMullen, Hausdorff dimension and conformal dynamics. Ⅲ. Computation of dimension, Amer. J. Math., 120 (1998), 691-721. doi: 10.1353/ajm.1998.0031. [6] C. McMullen, A. Mohammadi and H. Oh, Geodesic planes in hyperbolic 3-manifolds, H. Invent. Math. , (2016). Available from: http://gauss.math.yale.edu/~ho2/. [7] A. Mohammadi, H. Oh, Matrix coefficients, counting and primes for orbits of geometrically finite groups, J. Eur. Math. Soc.(JEMS), 17 (2015), 837-897. doi: 10.4171/JEMS/520. [8] A. Mohammadi, H. Oh, Ergodicity of unipotent flows and Kleinian groups, J. Amer. Math. Soc., 28 (2015), 531-577. doi: 10.1090/S0894-0347-2014-00811-0. [9] D. Mumford, C. Series, D. Wright, Indra's Pearls. The Vision of Felix Klein, Cambridge University Press, New York, 2002. doi: 10.1017/CBO9781107050051.024. [10] H. Oh, Harmonic analysis, ergodic theory and counting for thin groups, in Thin Groups and Superstrong Approximation (eds. E. Breulliard and H. Oh), Math. Sci. Res. Inst. Publ. , 61, Cambridge Univ. Press. , Cambridge, 2014. [11] H. Oh, N. Shah, Equidistribution and counting for orbits of geometrically finite hyper bolic groups, J. Amer. Math. Soc., 26 (2013), 511-562. doi: 10.1090/S0894-0347-2012-00749-8. [12] H. Oh, N. Shah, The asymptotic distribution of circles in the orbits of Kleinian groups, Invent. Math., 187 (2012), 1-35. doi: 10.1007/s00222-011-0326-7. [13] J. Parkkonen and F. Paulin, Counting common perpendicular arcs in negative curvature, To appear in Ergodic Theory and Dynamical Systems, arXiv: 1305.1332. [14] P. Vesselin, L. Stoyanov, Distribution of periods of closed trajectories in exponentially shrinking intervals, Comm. Math. Phys., 310 (2012), 675-704. doi: 10.1007/s00220-012-1419-x. [15] B. Stratmann, Urbański M., Diophantine extremality of the Patterson measure, Math. Proc. Cambridge Philos. Soc., 140 (2006), 297-304. doi: 10.1017/S0305004105009114. [16] B. Stratmann, S. L. Velani, The Patterson measure for geometrically finite groups with parabolic elements. new and old, Proc. London Math. Soc. (3), 71 (1995), 197-220. doi: 10.1112/plms/s3-71.1.197. [17] 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. [18] L. Stoyanov, Spectra of Ruelle transfer operators for axiom A flows, Nonlinearity, 24 (2011), 1089-1120. doi: 10.1088/0951-7715/24/4/005. [19] I. Vinogradov, Effective bisector estimate with applications to Apollonian circle packings, Int. Math. Res. Not. IMRN 2014,3217–3262.
Circle packing intersecting bounded region (background pictures are reproduced from Indra's Pearls: The Vision of Felix Klein, by D. Mumford, C. Series and D. Wright, copyright Cambridge University Press 2002)
Circle packing in ideal hyperbolic triangle
Apollonian circle packing $\mathscr{P}$ and generators of $\mathcal{A}$
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