Counting common perpendicular arcs in negative curvature

Abstract

Let $D^{-}$ and $D^{+}$ be properly immersed closed locally convex subsets of a Riemannian manifold with pinched negative sectional curvature. Using mixing properties of the geodesic flow, we give an asymptotic formula as $t\to +\infty$ for the number of common perpendiculars of length at most $t$ from $D^{-}$ to $D^{+}$, counted with multiplicities, and we prove the equidistribution in the outer and inner unit normal bundles of $D^{-}$ and $D^{+}$ of the tangent vectors at the endpoints of the common perpendiculars. When the manifold is compact with exponential decay of correlations or arithmetic with finite volume, we give an error term for the asymptotic. As an application, we give an asymptotic formula for the number of connected components of the domain of discontinuity of Kleinian groups as their diameter goes to $0$.

38 pages. Major revision (20 pages shorter). Equidistribution of equidistant surfaces and the case with potential and Gibbs measures moved to book in preparation [BPP]

References in corpus (5)

• Equidistribution and Counting for orbits of geometrically finite hyperbolic groups
• The Asymptotic distribution of circles in the orbits of Kleinian groups
• Counting visible circles on the sphere and Kleinian groups
• Skinning measures in negative curvature and equidistribution of equidistant submanifolds
• A relative trace formula for a compact Riemann surface

Cited by in corpus (1)

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