American Institute 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:

show all references

References:
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 $\mathcal{A}$
 [1] Joachim Escher, Boris Kolev. Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle. Journal of Geometric Mechanics, 2014, 6 (3) : 335-372. doi: 10.3934/jgm.2014.6.335 [2] Yury Neretin. The group of diffeomorphisms of the circle: Reproducing kernels and analogs of spherical functions. Journal of Geometric Mechanics, 2017, 9 (2) : 207-225. doi: 10.3934/jgm.2017009 [3] Or Landesberg. Horospherically invariant measures and finitely generated Kleinian groups. Journal of Modern Dynamics, 2021, 17: 337-352. doi: 10.3934/jmd.2021012 [4] A. A. Pinto, D. Sullivan. The circle and the solenoid. Discrete & Continuous Dynamical Systems, 2006, 16 (2) : 463-504. doi: 10.3934/dcds.2006.16.463 [5] S. R. Bullett and W. J. Harvey. Mating quadratic maps with Kleinian groups via quasiconformal surgery. Electronic Research Announcements, 2000, 6: 21-30. [6] Chuanxin Zhao, Lin Jiang, Kok Lay Teo. A hybrid chaos firefly algorithm for three-dimensional irregular packing problem. Journal of Industrial & Management Optimization, 2020, 16 (1) : 409-429. doi: 10.3934/jimo.2018160 [7] Wenxun Xing, Feng Chen. A-shaped bin packing: Worst case analysis via simulation. Journal of Industrial & Management Optimization, 2005, 1 (3) : 323-335. doi: 10.3934/jimo.2005.1.323 [8] Shinji Imahori, Yoshiyuki Karuno, Kenju Tateishi. Pseudo-polynomial time algorithms for combinatorial food mixture packing problems. Journal of Industrial & Management Optimization, 2016, 12 (3) : 1057-1073. doi: 10.3934/jimo.2016.12.1057 [9] Jimmy Tseng. On circle rotations and the shrinking target properties. Discrete & Continuous Dynamical Systems, 2008, 20 (4) : 1111-1122. doi: 10.3934/dcds.2008.20.1111 [10] Hicham Zmarrou, Ale Jan Homburg. Dynamics and bifurcations of random circle diffeomorphism. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 719-731. doi: 10.3934/dcdsb.2008.10.719 [11] Heather Hannah, A. Alexandrou Himonas, Gerson Petronilho. Anisotropic Gevrey regularity for mKdV on the circle. Conference Publications, 2011, 2011 (Special) : 634-642. doi: 10.3934/proc.2011.2011.634 [12] Carlos Gutierrez, Simon Lloyd, Vladislav Medvedev, Benito Pires, Evgeny Zhuzhoma. Transitive circle exchange transformations with flips. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 251-263. doi: 10.3934/dcds.2010.26.251 [13] Mao Chen, Xiangyang Tang, Zhizhong Zeng, Sanya Liu. An efficient heuristic algorithm for two-dimensional rectangular packing problem with central rectangle. Journal of Industrial & Management Optimization, 2020, 16 (1) : 495-510. doi: 10.3934/jimo.2018164 [14] Rafael De La Llave, Michael Shub, Carles Simó. Entropy estimates for a family of expanding maps of the circle. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 597-608. doi: 10.3934/dcdsb.2008.10.597 [15] Alena Erchenko. Flexibility of Lyapunov exponents for expanding circle maps. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2325-2342. doi: 10.3934/dcds.2019098 [16] Liviana Palmisano. Unbounded regime for circle maps with a flat interval. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 2099-2122. doi: 10.3934/dcds.2015.35.2099 [17] Abdumajid Begmatov, Akhtam Dzhalilov, Dieter Mayer. Renormalizations of circle hoemomorphisms with a single break point. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4487-4513. doi: 10.3934/dcds.2014.34.4487 [18] Saša Kocić, João Lopes Dias. Reducibility of quasi-periodically forced circle flows. Discrete & Continuous Dynamical Systems, 2020, 40 (9) : 5325-5345. doi: 10.3934/dcds.2020229 [19] Yves Cornulier. Realizations of groups of piecewise continuous transformations of the circle. Journal of Modern Dynamics, 2020, 16: 59-80. doi: 10.3934/jmd.2020003 [20] Wen Huang, Jianya Liu, Ke Wang. Möbius disjointness for skew products on a circle and a nilmanifold. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3531-3553. doi: 10.3934/dcds.2021006

2020 Impact Factor: 0.848