Logarithm laws for strong unstable foliations in negative curvature and non-Archimedian Diophantine approximation

Given a finite volume negatively curved Riemannian manifoldM, we give a precise relation between the logarithmic growth rates of the excursions of the strong unstable leaves of negatively recurrent unit tangent vectors into cusp neighborhoods ofM and their linear divergence rates under the geodesic flow. Our results hold in the more general setting whereM is the quotient of any proper CAT.-1/ metric space X by any geometrically finite discrete group of isometries of X. As an application to non-Archimedian Diophantine approximation in positive characteristic, we relate the growth of the orbits of ∂_K̂ -lattices under one-parameter unipotent subgroups of GL₂ (K̂) with approximation exponents and continued fraction expansions of elements of the local field K̂ of formal Laurent series over a finite field.