Density and Equidistribution of One-Sided Horocycles of a Geometrically Finite Hyperbolic Surface

On geometrically finite negatively curved surfaces, we give necessary and sufficient conditions for a one-sided horocycle $(h^s u)_{s\ge 0}$ to be dense in the nonwandering set of the geodesic flow. We prove that all dense one-sided orbits $(h^su)_{s\ge 0}$ are equidistributed, extending results of [Bu] and [Scha2] where symmetric horocycles $(h^su)_{s\in\R}$ were considered.Comment: 15 page

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 $3$ manifolds are related. Our results apply to Apollonian circle packings, Sierpinski curves, Schottky dances, etc.Comment: To appear in the Proceedings of ICM, 201

Distribution of orbits in R2\R^2 of a finitely generated group of \SL(2,\R)

In this work, we study the asymptotic distribution of the non discrete orbits of a finitely generated group acting linearly on $\R^2$. To do this, we establish new equidistribution results for the horocyclic flow on the unitary tangent bundle of the associated surface

Homogeneous Dynamics and Number Theory

The theory of flows on homogeneous spaces of Lie groups has emerged as a distinct, rapidly advancing subject over the last few decades incorporating ergodic theory, geometry and number theory. The workshop showcased the latest advances in the subject as well as a wide range of applications

Equidistribution and Counting for orbits of geometrically finite hyperbolic groups

Let G be the identity component of SO(n,1), acting linearly on a finite dimensional real vector space V. Consider a vector w_0 in V such that the stabilizer of w_0 is a symmetric subgroup of G or the stabilizer of the line Rw_0 is a parabolic subgroup of G. For any non-elementary discrete subgroup Gamma of G with w_0Gamma discrete, we compute an asymptotic formula for the number of points in w_0Gamma of norm at most T, provided that the Bowen-Margulis-Sullivan measure on the associated hyperbolic manifold and the Gamma skinning size of w_0 are finite. The main ergodic ingredient in our approach is the description for the limiting distribution of the orthogonal translates of a totally geodesically immersed closed submanifold of Gamma\H^n. We also give a criterion on the finiteness of the Gamma skinning size of w_0 for Gamma geometrically finite.Comment: Extensions of Equidistribution results to G/Gamma are obtained for Gamma Zariski dense, and Much more precise description on the structure of cuspidal neighborhoods of parabolic points is obtained for Gamma geometrically finite. 63 pages (with 1 figure