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The asymptotic distribution of circles in the orbits of Kleinian groups

The asymptotic distribution of circles in the orbits of Kleinian groups,10.1007/s00222-011-0326-7,Inventiones Mathematicae,Hee Oh,Nimish Shah

The asymptotic distribution of circles in the orbits of Kleinian groups   (Citations: 2)
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Let 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 , assuming that either the critical exponent of Γ is strictly bigger than 1 or does not contain an infinite bouquet of tangent circles glued at a parabolic fixed point of Γ. Our construction also works for invariant under a geometrically infinite group Γ, provided Γ admits a finite Bowen-Margulis-Sullivan measure and the Γ-skinning size of is finite. Some concrete circle packings to which our result applies include Apollonian circle packings, Sierpinski curves, Schottky dances, etc.
Journal: Inventiones Mathematicae - INVENT MATH , vol. 187, no. 1, pp. 1-3, pp. 1-35, 2012
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