Journal of Modern Dynamics (JMD)

Invariant distributions for homogeneous flows and affine transformations

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

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Livio Flaminio - UMR CNRS 8524, UFR de Mathématiques, Université de Lille 1, F59655 Villeneuve d’Asq CEDEX, France (email)
Giovanni Forni - Department of Mathematics, University of Maryland, College Park, MD 20742-4015, United States (email)
Federico Rodriguez Hertz - Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, United States (email)

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.

Received: May 2013;      Revised: December 2015;      Available Online: March 22 2016.