Dynamics on geometrically finite hyperbolic manifolds with applications to Apollonian circle packings and beyond

TL;DR

This paper explores the dynamics of flows on geometrically finite hyperbolic 3-manifolds and applies these insights to count and analyze circle packings like Apollonian packings, revealing new connections and results.

Contribution

It introduces new methods linking hyperbolic dynamics with circle packing distributions, extending understanding to various fractal and geometric structures.

Findings

Counting and distribution results for circles in packings

Connections between hyperbolic flow dynamics and circle packings

Applications to Apollonian, Sierpinski, and Schottky structures

Abstract

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 $3$ manifolds are related. Our results apply to Apollonian circle packings, Sierpinski curves, Schottky dances, etc.