Journal of Modern Dynamics (JMD)

Effective equidistribution of circles in the limit sets of Kleinian groups

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

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Wenyu Pan - Mathematics Department, Yale University, New Haven, CT 06520, United States (email)

Abstract: Consider a general circle packing $\mathfrak{P}$ 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 $\mathfrak{P}$ 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 $\mathfrak{P}(\mathfrak{T})$ 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.

Received: November 2015;      Revised: October 2016;      Available Online: February 2017.