8,393 research outputs found
Apollonian circle packings: Dynamics and Number theory
We give an overview of various counting problems for Apollonian circle
packings, which turn out to be related to problems in dynamics and number
theory for thin groups. This survey article is an expanded version of my
lecture notes prepared for the 13th Takagi lectures given at RIMS, Kyoto in the
fall of 2013.Comment: To appear in Japanese Journal of Mat
Dynamics on geometrically finite hyperbolic manifolds with applications to Apollonian circle packings and beyond
We present recent results on counting and distribution of circles in a given
circle packing invariant under a geometrically finite Kleinian group and
discuss how the dynamics of flows on geometrically finite hyperbolic
manifolds are related. Our results apply to Apollonian circle packings,
Sierpinski curves, Schottky dances, etc.Comment: To appear in the Proceedings of ICM, 201
Effective Circle Count for Apollonian packings and Closed horospheres
The main result of this paper is an effective count for Apollonian circle
packings that are either bounded or contain two parallel lines. We obtain this
by proving an effective equidistribution of closed horospheres in the unit
tangent bundle of a geometrically finite hyperbolic 3-manifold of infinite
volume, whose fundamental group has critical exponent bigger than 1. We also
discuss applications to Affine sieves. Analogous results for surfaces are
treated as well.Comment: 43 pages, 2 figures, To appear in GAF
Counting visible circles on the sphere and Kleinian groups
For a circle packing P on the sphere invariant under a geometrically finite
Kleinian group, we compute the asymptotic of the number of circles in P of
spherical curvature at most which are contained in any given region.Comment: Main results are significantly improved, 16 pages, 1 figur
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