Geodesic
Logarithm laws for strong unstable foliations in negative curvature and non-Archimedian Diophantine approximation
Jayadev S. Athreya1 orcid ; Frédéric Paulin2
1University of Illinois at Urbana-Champaign, USA
2Ecole Normale Superieure, Paris, France
Groups, geometry, and dynamics, Tome 8 (2014) no. 2, pp. 285-309

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DOI

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 OK^​-lattices under one-parameter unipotent subgroups of GL2​(K^) with approximation exponents and continued fraction expansions of elements of the local field K^ of formal Laurent series over a finite field.
DOI : 10.4171/ggd/226
Classification : 37-XX
Mots-clés : Negative curvature, geodesic flow, horocyclic flow, strong unstable foliation, cusp excursions, logarithm law, Diophantine approximation, continued fraction, approximation exponent

Jayadev S. Athreya  1   ; Frédéric Paulin  2

1 University of Illinois at Urbana-Champaign, USA
2 Ecole Normale Superieure, Paris, France 
Jayadev S. Athreya; Frédéric Paulin. Logarithm laws for strong unstable foliations in negative curvature and non-Archimedian Diophantine approximation. Groups, geometry, and dynamics, Tome 8 (2014) no. 2, pp. 285-309. doi: 10.4171/ggd/226
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     author = {Jayadev S. Athreya and Fr\'ed\'eric Paulin},
     title = {Logarithm laws for strong unstable foliations in negative curvature and {non-Archimedian} {Diophantine} approximation},
     journal = {Groups, geometry, and dynamics},
     pages = {285--309},
     year = {2014},
     volume = {8},
     number = {2},
     doi = {10.4171/ggd/226},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/226/}
}
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