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On the rate of equidistribution of expanding horospheres in finite-volume quotients of SL(2, ${\mathbb{C}}$)
2017, 11: 189-217. doi: 10.3934/jmd.2017009
Effective equidistribution of circles in the limit sets of Kleinian groups
Wenyu Pan
Mathematics Department, Yale University, New Haven, CT 06520, USA
Received November 25, 2015 Revised October 22, 2016 Published February 2017
Figure(3)
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].
Keywords: Circle packing, Kleinian group.
Mathematics Subject Classification: Primary: 37A17; Secondary: 30F40, 52C26.
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
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Figure 1. 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)
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Figure 2. Circle packing in ideal hyperbolic triangle
Figure 3. Apollonian circle packing $\mathscr{P}$ and generators of $\mathcal{A}$
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