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Inventiones mathematicae

, Volume 187, Issue 1, pp 1–35 | Cite as

The asymptotic distribution of circles in the orbits of Kleinian groups

  • Hee Oh
  • Nimish Shah
Article

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.

Keywords

Asymptotic Distribution Borel Subset Kleinian Group Circle Packing Schottky Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Mathematics DepartmentBrown UniversityProvidenceUSA
  2. 2.Korea Institute for Advanced StudySeoulKorea
  3. 3.Department of MathematicsThe Ohio State UniversityColumbusUSA

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