Given a finite volume negatively curved Riemannian manifold , 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 and their linear divergence rates under the geodesic flow. Our results hold in the more general setting where is the quotient of any proper CAT(±1) metric space by any geometrically finite discrete group of isometries of . 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 with approximation exponents and continued fraction expansions of elements of the local field of formal Laurent series over a finite field.
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Jayadev S. Athreya, Frédéric Paulin, Logarithm laws for strong unstable foliations in negative curvature and non-Archimedian Diophantine approximation. Groups Geom. Dyn. 8 (2014), no. 2, pp. 285–309DOI 10.4171/GGD/226