2 research outputs found
The Asymptotic distribution of circles in the orbits of Kleinian groups
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.Comment: 31 pages, 8 figures. Final version. To appear in Inventiones Mat