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

Abstract

Given a finite volume negatively curved Riemannian manifold $M$, 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 of $M$ and their linear divergence rates under the geodesic flow. Our results hold in the more general setting where $M$ 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 $\mathcal{O}\_{\hat K}$-lattices under one-parameter unipotent subgroups of $\mathrm{GL}\_2(\hat K)$ with approximation exponents and continued fraction expansions of elements of the local field $\hat K$ of formal Laurent series over a finite field.