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

Jayadev S. Athreya (1) , Frédéric Paulin (2)

1 Department of Mathematics [Urbana]
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay

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.

Mots clés: negative curvature; geodesic flow; horocyclic flow; strong unstable foliation; cusp excursions; logarithm law; Diophantine approximation; continued fraction; approximation exponent

Domaines: Systèmes dynamiques [math.DS]; Géométrie différentielle [math.DG]; Théorie des nombres [math.NT]; Théorie des groupes [math.GR]

Jayadev S. Athreya, Frédéric Paulin. Logarithm laws for strong unstable foliations in negative curvature and non-Archimedian Diophantine approximation. 2012. ⟨hal-00699060⟩