# Groups, Geometry, and Dynamics

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**Volume 8, Issue 2, 2014, pp. 285–309**

**DOI: 10.4171/GGD/226**

Published online: 2014-07-08

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

Jayadev S. Athreya^{[1]}and Frédéric Paulin

^{[2]}(1) University of Illinois at Urbana-Champaign, USA

(2) Ecole Normale Superieure, Paris, France

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.

*Keywords: *Negative curvature, geodesic flow, horocyclic flow, strong unstable foliation, cusp excursions, logarithm law, Diophantine approximation, continued fraction, approximation exponent

Athreya Jayadev, Paulin Frédéric: Logarithm laws for strong unstable foliations in negative curvature and non-Archimedian Diophantine approximation. *Groups Geom. Dyn.* 8 (2014), 285-309. doi: 10.4171/GGD/226