Ultrametric Logarithm Laws, II

We prove positive characteristic versions of the logarithm laws of Sullivan and Kleinbock-Margulis and obtain related results in Metric Diophantine Approximation.Comment: submitted to Montasefte Fur Mathemati

Khinchin theorem for integral points on quadratic varieties

We prove an analogue the Khinchin theorem for the Diophantine approximation by integer vectors lying on a quadratic variety. The proof is based on the study of a dynamical system on a homogeneous space of the orthogonal group. We show that in this system, generic trajectories visit a family of shrinking subsets infinitely often.Comment: 19 page

Length functions on currents and applications to dynamics and counting

The aim of this (mostly expository) article is twofold. We first explore a variety of length functions on the space of currents, and we survey recent work regarding applications of length functions to counting problems. Secondly, we use length functions to provide a proof of a folklore theorem which states that pseudo-Anosov homeomorphisms of closed hyperbolic surfaces act on the space of projective geodesic currents with uniform north-south dynamics.Comment: 35pp, 2 figures, comments welcome! Second version: minor corrections. To appear as a chapter in the forthcoming book "In the tradition of Thurston" edited by V. Alberge, K. Ohshika and A. Papadopoulo