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Joint partial equidistribution of Farey rays in negatively curved manifolds and trees

Part of: Multiplicative number theory Dynamical systems with hyperbolic behavior Structure and classification of infinite or finite groups Real and complex geometry Linear algebraic groups and related topics Classical measure theory Global differential geometry

Published online by Cambridge University Press:  08 January 2024

JOUNI PARKKONEN  and
FRÉDÉRIC PAULIN
JOUNI PARKKONEN
Affiliation:
Department of Mathematics and Statistics, P.O. Box 35, 40014 University of Jyväskylä, Finland (e-mail: jouni.t.parkkonen@jyu.fi)
FRÉDÉRIC PAULIN*
Affiliation:
Laboratoire de mathématique d’Orsay, UMR 8628 CNRS, Université Paris-Saclay, 91405 Orsay Cedex, France
*
e-mail: frederic.paulin@universite-paris-saclay.fr
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Abstract

We prove a joint partial equidistribution result for common perpendiculars with given density on equidistributing equidistant hypersurfaces, towards a measure supported on truncated stable leaves. We recover a result of Marklof on the joint partial equidistribution of Farey fractions at a given density, and give several analogous arithmetic applications, including in Bruhat–Tits trees.

Keywords

equidistributionnegative curvaturegeodesic flowcommon perpendicularsFarey fractions

MSC classification

Primary: 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 53C22: Geodesics 11N45: Asymptotic results on counting functions for algebraic and topological structures
Secondary: 20G20: Linear algebraic groups over the reals, the complexes, the quaternions 28A33: Spaces of measures, convergence of measures 51M10: Hyperbolic and elliptic geometries (general) and generalizations 20E08: Groups acting on trees
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 44 , Issue 9 , September 2024 , pp. 2700 - 2736
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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