On measures invariant under diagonalizable actions: the rank-one case and the general low-entropy method
type: Item
instance of: scholarly article
title: On measures invariant under diagonalizable actions: the rank-one case and the general low-entropy method
zbMATH Open document ID: 1146.37006
author: Manfred Einsiedler
author: Elon Lindenstrauss
published in: Journal of Modern Dynamics
publication date: 2008-06-18T00:00:00Z
review text: Many important problems in dynamical systems and its interactions with number theory involve classifying measures invariant under the action of a real-diagonalizable group on some locally compact quotient of a real algebraic group. All the results in this spirit have involved some positive entropy hypothesis, and some of the results have required other mixing hypotheses on the measure. Two important steps in the direction of controlling the hypotheses needed are a ``high-entropy'' method, introduced by \textit{M. Einsiedler} and \textit{A. Katok} [Isr. J. Math. 148, 169--238 (2005; Zbl 1097.37017)] (which requires a genuine action of a multi-dimensional group and more than simply positive entropy for a single element) and a ``low-entropy'' method, introduced by \textit{E. Lindenstrauss} [Ann. Math. (2) 163, No. 1, 165--219 (2006; Zbl 1104.22015)] (in which information about the measure is extracted from properties of a single transformation). Because a typical situation involves a completely unknown measure, these two approaches are complementary, and indeed were both brought to bear in the work of the authors and \textit{A. Katok} on Littlewood's problem [Ann. Math. (2) 164, No. 2, 513--560 (2006; Zbl 1109.22004)]. In this paper the low-entropy method is substantially generalized, with the development of rigidity properties for the measurable factors of the action of a single positive entropy element (for an unknown measure) on certain locally homogeneous spaces \(\Lambda\backslash G\). This is used to classify positive entropy measures invariant under a one-parameter group with an extra recurrence condition when \(G\) is a product with a rank-one algebraic group.
Mathematics Subject Classification ID: 37A35
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 37A45
Mathematics Subject Classification ID: 11H46
Mathematics Subject Classification ID: 22D40
Mathematics Subject Classification ID: 28D20
zbMATH DE Number: 5290036
zbMATH Keywords: torus action
zbMATH Keywords: homogeneous space
zbMATH Keywords: invariant measure
zbMATH Keywords: entropy
reviewed by: Thomas B. Ward
MaRDI profile type: Publication
Recommended article: Q5490972
Recommended article: SOME EXAMPLES HOWTO USE MEASURE CLASSIFICATION IN NUMBER THEORY
Recommended article: On measures invariant under tori on quotients of semisimple groups
Recommended article: Diagonal actions in positive characteristic
Recommended article: Joinings of higher-rank diagonalizable actions on locally homogeneous spaces
Recommended article: Exposé Bourbaki 1185 : Joinings classification and applications (after Einsiedler and Lindenstrauss)
Recommended article: Possible entropy functions
Recommended article: Rigidity properties of ℤ^{𝕕}-actions on tori and solenoids
Recommended article: Invariant measures for actions of higher rank abelian groups
Recommended article: Algebraic $\mathbb {Z}^d$-actions of entropy rank one
description: scientific article; zbMATH DE number 5290036