The Asymptotic distribution of circles in the orbits of Kleinian groups

TL;DR

This paper studies the distribution of small circles in invariant packings under Kleinian groups, providing explicit measures and extending results to geometrically infinite groups with certain conditions.

Contribution

It constructs explicit measures describing circle distribution in Kleinian group orbits, extending previous results to geometrically infinite groups with finite Bowen-Margulis-Sullivan measure.

Findings

Explicit measure for circle distribution in geometrically finite groups

Extension of results to geometrically infinite groups under certain conditions

Application to classical circle packings like Apollonian and Sierpinski

Abstract

Let P be a locally finite circle packing in the plane invariant under a non-elementary Kleinian group Gamma and with finitely many Gamma-orbits. When Gamma is geometrically finite, we construct an explicit Borel measure on the plane which describes the asymptotic distribution of small circles in P, assuming that either the critical exponent of Gamma is strictly bigger than 1 or P does not contain an infinite bouquet of tangent circles glued at a parabolic fixed point of Gamma. Our construction also works for P invariant under a geometrically infinite group Gamma, provided Gamma admits a finite Bowen-Margulis-Sullivan measure and the Gamma-skinning size of P is finite. Some concrete circle packings to which our result applies include Apollonian circle packings, Sierpinski curves, Schottky dances, etc.