Equidistribution of common perpendiculars in negative curvature

TL;DR

This paper proves that measures supported on common perpendiculars between convex subsets in negatively curved manifolds become uniformly distributed according to the Bowen-Margulis measure as the length grows, with error estimates in certain cases.

Contribution

It establishes equidistribution results for common perpendiculars in negatively curved manifolds, including error terms and weighted versions with potentials.

Findings

Measures on common perpendiculars equidistribute to Bowen-Margulis measure as length increases.

Error terms are provided for locally symmetric finite volume manifolds with exponential mixing.

Weighted equidistribution holds when incorporating bounded H"older potentials and their equilibrium states.

Abstract

Let $A^-$ and $A^+$ be properly immersed closed locally convex subsets of a Riemannian manifold $M$ with pinched negative sectional curvature. When the Bowen-Margulis measure on $T^1M$ is finite and mixing for the geodesic flow, we prove that the Lebesgue measures along the common perpendiculars of length at most $t$ from $A^-$ to $A^+$, counted with multiplicities and lifted to $T^1M$, equidistribute to the Bowen-Margulis measure as $t\to+\infty$. When $M$ is locally symmetric with finite volume and the geodesic flow is exponentially mixing, we give an error term for the asymptotic. When $T^1M$ is endowed with a bounded H\"older-continuous potential, and when the associated equilibrium state is finite and mixing for the geodesic flow, we prove the equidistribution of these Lebesgue measures weighted by the amplitudes of the potential to the equilibrium state.