The asymptotic distribution of circles in the orbits of Kleinian groups

• 成果类型: Article
• 署名作者: Oh, Hee; Shah, Nimish
• 署名单位: Brown University; Korea Institute for Advanced Study (KIAS); University System of Ohio; Ohio State University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-011-0326-7
• 发表日期: 2012
• 页码: 1-35
• 关键词: dimension; packings; points; set

摘要

Let P be a locally finite circle packing in the plane C 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 C 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.