Unique ergodicity of the horocycle flow on Riemannnian foliations

TL;DR

This paper proves that the horocycle flow on Riemannian foliations is uniquely ergodic and minimal, extending classical results from hyperbolic surfaces to a broader foliated setting, confirming a conjecture by Matsumoto.

Contribution

It establishes the strict ergodicity and minimality of the horocycle flow on Riemannian foliations, using Coudène's theorem, thus generalizing known results beyond hyperbolic surfaces.

Findings

Horocycle flow is strictly ergodic on Riemannian foliations.

The flow is minimal, confirming Matsumoto's conjecture.

Application of Coudène's theorem to foliated spaces.

Abstract

A classic result due to Furstenberg is the strict ergodicity of the horocycle flow for a compact hyperbolic surface. Strict ergodicity is unique ergodicity with respect to a measure of full support, and therefore implies minimality. The horocycle flow has been previously studied on minimal foliations by hyperbolic surfaces on closed manifolds, where it is known not to be minimal in general. In this paper, we prove that for the special case of Riemannian foliations, strict ergodicity of the horocycle flow still holds. This in particular proves that this flow is minimal, which establishes a conjecture proposed by Matsumoto. The main tool is a theorem due to Coud\`ene, which he presented as an alternative proof for the surface case. It applies to two continuous flows defining a measure-preserving action of the affine group of the line on a compact metric space, precisely matching the foliated setting. In addition, we briefly discuss the application of Coud\`ene's theorem to other kinds of foliations.