Invariant distributions for homogeneous flows and affine transformations

Livio  Flaminio; Giovanni Forni; Federico Rodriguez Hertz

doi:10.3934/jmd.2016.10.33

Article Contents

2016, Volume 10: 33-79. Doi: 10.3934/jmd.2016.10.33

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Invariant distributions for homogeneous flows and affine transformations

1.
UMR CNRS 8524, UFR de Mathématiques, Université de Lille 1, F59655 Villeneuve d’Asq CEDEX
2.
Department of Mathematics, University of Maryland, College Park, MD 20742-4015
3.
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802

Received: April 30, 2013

Revised: November 30, 2015

Published: March 2016

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Abstract

We prove that every homogeneous flow on a finite-volume homogeneous manifold has countably many independent invariant distributions unless it is conjugate to a linear flow on a torus. We also prove that the same conclusion holds for every affine transformation of a homogenous space which is not conjugate to a toral translation. As a part of the proof, we have that any smooth partially hyperbolic flow on any compact manifold has countably many distinct minimal sets, hence countably many distinct ergodic probability measures. As a consequence, the Katok and Greenfield-Wallach conjectures hold in all of the above cases.

Keywords:

Cohomological equations,
homogeneous flows.

Mathematics Subject Classification: Primary: 37-XX, 37C15, 37C40.

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