Journal of Modern Dynamics (JMD)

On measures invariant under diagonalizable actions: the Rank-One case and the general Low-Entropy method

Pages: 83 - 128, Issue 1, January 2008      doi:10.3934/jmd.2008.2.83

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Manfred Einsiedler - Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus OH 43210-1174, United States (email)
Elon Lindenstrauss - Fine Hall, Washington Road, Princeton NJ 08544-1000, United States (email)

Abstract: We consider measures on locally homogeneous spaces $\Gamma \backslash G$ which are invariant and have positive entropy with respect to the action of a single diagonalizable element $a \in G$ by translations, and prove a rigidity statement regarding a certain type of measurable factors of this action.
    This rigidity theorem, which is a generalized and more conceptual form of the low entropy method of [14,3] is used to classify positive entropy measures invariant under a one parameter group with an additional recurrence condition for $G=G_1 \times G_2$ with $G_1$ a rank one algebraic group. Further applications of this rigidity statement will appear in forthcoming papers.

Keywords:  torus action, homogeneous space, invariant measures, entropy.
Mathematics Subject Classification:  Primary: 37D40; Secondary: 37A35, 37A45.

Received: August 2007;      Revised: October 2007;      Available Online: October 2007.