Counting arcs in negative curvature

2012/03/01 by Jouni Parkkonen, Parkkonen, Jouni, Frédéric Paulin +1 · 2 citations
Mathematics · #11E39 #11N45 #11R52 #20G20 #20H10 #30F40 #37A25 #37D40 #53C22 #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT) #math.DG #math.DS #math.NT #msc:11E39 #msc:11N45 #msc:11R52 #msc:20G20 #msc:20H10 #msc:30F40 #msc:37A25 #msc:37D40 #msc:53C22

paper · pdf · doi:10.48550/arxiv.1203.0175

Revised version, 44 pages

arxiv created 2013/05/07 · arxiv updated 2013/05/09

Abstract

Let M be a complete Riemannian manifold with negative curvature, and let C_-, C₊ be two properly immersed closed convex subsets of M. We survey the asymptotic behaviour of the number of common perpendiculars of length at most s from C_- to C₊, giving error terms and counting with weights, starting from the work of Huber, Herrmann, Margulis and ending with the works of the authors. We describe the relationship with counting problems in circle packings of Kontorovich, Oh, Shah. We survey the tools used to obtain the precise asymptotics (Bowen-Margulis and Gibbs measures, skinning measures). We describe several arithmetic applications, in particular the ones by the authors on the asymptotics of the number of representations of integers by binary quadratic, Hermitian or Hamiltonian forms.

Cited by

• Hausdorff Dimension of non-conical and Myrberg limit sets
• Counting and equidistribution of strongly reversible closed geodesics in negative curvature

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