# American Institute of Mathematical Sciences

January  2008, 2(1): 83-128. doi: 10.3934/jmd.2008.2.83

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

 1 Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus OH 43210-1174, United States 2 Fine Hall, Washington Road, Princeton NJ 08544-1000, United States

Received  August 2007 Revised  October 2007 Published  October 2007

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.
Citation: Manfred Einsiedler, Elon Lindenstrauss. On measures invariant under diagonalizable actions: the Rank-One case and the general Low-Entropy method. Journal of Modern Dynamics, 2008, 2 (1) : 83-128. doi: 10.3934/jmd.2008.2.83
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