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Equidistribution of common perpendiculars in negative curvature

Jouni Parkkonen, Frédéric Paulin

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 Hölder-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.

Equidistribution of common perpendiculars in negative curvature

Abstract

Let and be properly immersed closed locally convex subsets of a Riemannian manifold with pinched negative sectional curvature. When the Bowen-Margulis measure on is finite and mixing for the geodesic flow, we prove that the Lebesgue measures along the common perpendiculars of length at most from to , counted with multiplicities and lifted to , equidistribute to the Bowen-Margulis measure as . When is locally symmetric with finite volume and the geodesic flow is exponentially mixing, we give an error term for the asymptotic. When is endowed with a bounded Hölder-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.

Paper Structure

This paper contains 6 sections, 4 theorems, 83 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The geometric and measure theoretic background
  3. Equidistribution of Lebesgue measures along common perpendiculars
  4. Potentials and equidistribution of common perpendiculars
  5. Potentials and Gibbs measures
  6. Proof of Theorem \ref{['theo:intropot']}

Key Result

Theorem 1

Assume that the Bowen-Margulis measure $m_{\rm BM}$ is finite and mixing for the geodesic flow of $M$. Assume that the skinning measures $\sigma^+_{A^-}$ and $\sigma^{-}_{A^+}$ are finite and nonzero. For the narrow convergence of measures on $T^1M$, we have If furthermore $M$ is locally symmetric with finite volume, and if the geodesic flow of $M$ is exponentially mixing, then there exists $\ell

Figures (2)

  • Figure 1: Geodesic loops equidistribute.
  • Figure 2: Splitting common perpendiculars.

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4