JournalsggdVol. 8, No. 2pp. 285–309

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

  • Jayadev S. Athreya

    University of Illinois at Urbana-Champaign, USA
  • Frédéric Paulin

    Ecole Normale Superieure, Paris, France
Logarithm laws for strong unstable foliations in negative curvature and non-Archimedian Diophantine approximation cover
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Abstract

Given a finite volume negatively curved Riemannian manifold MM, 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 MM and their linear divergence rates under the geodesic flow. Our results hold in the more general setting where MM is the quotient of any proper CAT(±1) metric space XX by any geometrically finite discrete group of isometries of XX. As an application to non-Archimedian Diophantine approximation in positive characteristic, we relate the growth of the orbits of OK^\mathcal{O}_{\hat K}-lattices under one-parameter unipotent subgroups of GL2(K^)\mathrm{GL}_2(\hat K) with approximation exponents and continued fraction expansions of elements of the local field K^\hat K of formal Laurent series over a finite field.

Cite this article

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. 285–309

DOI 10.4171/GGD/226