The Asymptotic distribution of circles in the orbits of Kleinian groups

Details The Asymptotic distribution of circles in the orbits of Kleinian groups The Asymptotic distribution of circles in the orbits of Kleinian groups

2010-04-13

arxiv.org/abs/1004.2130v2

Inventiones Math. 187 (2012), no. 1, 1-35

Mathematics

Dynamical Systems

31 pages, 8 figures. Final version. To appear in Inventiones Math

Scientific paper

10.1007/s00222-011-0326-7

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.

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