Logarithm laws for strong unstable foliations in negative curvature and   non-Archimedian Diophantine approximation
  Given for instance a finite volume negatively curved Riemannian manifold $M$, we give a precise relation between the logarithmic growth rates of the excursions into cusps neighborhoods of the strong unstable leaves of negatively recurrent unit vectors of $M$ and their linear divergence rates under the geodesic flow. As an application to non-Archimedian Diophantine approximation in positive characteristic, we relate the growth of the orbits of lattices under one-parameter unipotent subgroups of $\GL_2(\wh K)$ with approximation exponents and continued fraction expansions of elements of the field $\wh K$ of formal Laurent series over a finite field. 
Jayadev S. Athreya, Fr\'ed\'eric Paulin (LM-Orsay)
math.DS math.DG math.GR math.NT