The asymptotic distribution of circles in the orbits of Kleinian groups

Abstract

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