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arXiv · 1205.4515

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

Jayadev S. Athreya·Frédéric Paulin
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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.

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Jayadev S. Athreya, Frédéric Paulin. 2012-05-21. Logarithm laws for strong unstable foliations in negative curvature and non-Archimedian Diophantine approximation. https://arxiv.org/abs/1205.4515

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