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


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.


Asymptotic Distribution Borel Subset Kleinian Group Circle Packing Schottky Group 
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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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