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

TL;DR

This paper establishes a connection between the growth rates of geodesic excursions in negatively curved manifolds and Diophantine approximation in positive characteristic, linking geometric dynamics with number theory.

Contribution

It provides a precise relation between cusp excursion growth in negatively curved manifolds and orbit divergence, and applies this to non-Archimedean Diophantine approximation.

Findings

Logarithmic growth rates of geodesic excursions relate to linear divergence rates.

Growth of lattice orbits under unipotent flows connects to approximation exponents.

Results bridge geometric dynamics and Diophantine approximation in positive characteristic.

Abstract

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.