THE ASYMPTOTIC DISTRIBUTION OF CIRCLES IN THE ORBITS OF KLEINIAN GROUPS

Abstract

Let 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 cir-cles in P, assuming that either the critical exponent of Γ is strictly bigger than 1 or P does not contain an infinite bouquet of tangent cir-cles glued at a parabolic fixed point of Γ. Our construction also works for P invariant under a geometrically infinite group Γ, provided Γ ad-mits a finite Bowen-Margulis-Sullivan measure and the Γ-skinning size of P is finite. Some concrete circle packings to which our result applies include Apollonian circle packings, Sierpinski curves, Schottky dances, etc.