Unique ergodicity of the horocycle flow on Riemannnian foliations

Fernando Alcalde Cuesta, Françoise Dal'Bo, Matilde Martínez, Alberto Verjovsky

Published 2018-10-10Version 1

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.