Unique ergodicity of the horocycle flow on Riemannnian foliations
| Autor: | F. Alcalde Cuesta, Alberto Verjovsky, Matilde Martinez, Françoise Dal'Bo |
|---|---|
| Přispěvatelé: | Universidade de Santiago de Compostela [Spain] (USC ), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Instituto de Matemática y Estadística Rafael Laguardia [Montevideo] (IMERL), Universidad de la República [Montevideo] (UCUR), Instituto de Matematicas (UNAM), Universidad Nacional Autónoma de México (UNAM), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Universidad de la República [Montevideo] (UDELAR), Universidad Nacional Autónoma de México = National Autonomous University of Mexico (UNAM) |
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Pure mathematics
Mathematics::Dynamical Systems General Mathematics [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] Dynamical Systems (math.DS) 01 natural sciences Group action group actions 0103 physical sciences Affine group FOS: Mathematics 0101 mathematics Mathematics - Dynamical Systems classical ergodic theory Mathematics foliations Conjecture Applied Mathematics 010102 general mathematics Ergodicity 37C85 37D40 57R30 16. Peace & justice Surface (topology) Compact space Horocycle Flow (mathematics) horocycle flow 010307 mathematical physics Mathematics::Differential Geometry |
| Zdroj: | Ergodic Theory and Dynamical Systems Ergodic Theory and Dynamical Systems, Cambridge University Press (CUP), 2020, 40 (6), pp.1459-1479. ⟨10.1017/etds.2018.119⟩ Ergodic Theory and Dynamical Systems, 2020, 40 (6), pp.1459-1479. ⟨10.1017/etds.2018.119⟩ |
| ISSN: | 0143-3857 1469-4417 |
| DOI: | 10.1017/etds.2018.119⟩ |
| Popis: | 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. Comment: 18 pages, 1 figure, to appear in Ergodic Theory and Dynamical Systems |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|