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Unique ergodicity of the horocycle flow on Riemannnian foliations

Published online by Cambridge University Press:  13 November 2018

F. ALCALDE CUESTA
Affiliation:
Instituto de Matemáticas, Universidade de Santiago de Compostela, Rúa Lope Gómez de Marzoa s/n, E-15782 Santiago de Compostela, Spain email fernando.alcalde@usc.es
F. DAL’BO
Affiliation:
IRMAR, Bâtiment 22, Campus de Beaulieu, Université Rennes 1, 35042 Rennes Cedex, France email dalbo@univ-rennes1.fr
M. MARTÍNEZ
Affiliation:
IMERL, Facultad de Ingeniería Universidad de la República, Julio Herrera y Reissig 565, 11300 Montevideo, Uruguay email matildem@ng.edu.uy
A. VERJOVSKY
Affiliation:
Instituto de Matemáticas (Unidad Cuernavaca), UNAM. Av. Universidad s/n, Col. Lomas de Chamilpa CP 62210, Cuernavaca, Mexico email alberto@matcuer.unam.mx

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 it 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 Coudène, 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 Coudène’s theorem to other kinds of foliations.

Type
Original Article
Copyright
© Cambridge University Press, 2018

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