Ultrametric Logarithm Laws I

We announce ultrametric analogues of the results of Kleinbock-Margulis for shrinking target properties of semisimple group actions on symmetric spaces. The main applications are S-arithmetic Diophantine approximation results and logarithm laws for buildings, generalizing the work of Hersonsky-Paulin on trees.Comment: This announcement has been completely revised to reflect many new developments. Please direct all references to this NEW announcement. It is now co-authored work. Submitted to Discrete and Continuous Dynamical System

Escape of mass and entropy for diagonal flows in real rank one situations

Let $G$ be a connected semisimple Lie group of real rank 1 with finite center, let $\Gamma$ be a non-uniform lattice in $G$ and $a$ any diagonalizable element in $G$. We investigate the relation between the metric entropy of $a$ acting on the homogeneous space $\Gamma\backslash G$ and escape of mass. Moreover, we provide bounds on the escaping mass and, as an application, we show that the Hausdorff dimension of the set of orbits (under iteration of $a$) which miss a fixed open set is not full.Comment: 40 pages, 1 figur

The distribution of geodesic excursions into the neighborhood of a cone singularity on a hyperbolic 2-orbifold

A generic geodesic on a finite area, hyperbolic 2-orbifold exhibits an infinite sequence of penetrations into a neighborhood of a cone singularity, so that the sequence of depths of maximal penetration has a limiting distribution. The distribution function is the same for all such surfaces and is described by a fairly simple formula.Comment: 20 page

Logarithm laws and shrinking target properties

We survey some of the recent developments in the study of logarithm laws and shrinking target properties for various families of dynamical systems. We discuss connections to geometry, diophantine approximation, and probability theory.Comment: This is a survey paper written following the Conference on Measures and Dyanmics on groups and homogeneous spaces, at TIFR, Mumbai, in Dec. 2007. It is in honor of Prof. S.G. Dani's 60th Birthda