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Unique ergodicity on compact homogeneous spaces


Author: Barak Weiss
Journal: Proc. Amer. Math. Soc. 129 (2001), 585-592
MSC (1991): Primary 22F30
DOI: https://doi.org/10.1090/S0002-9939-00-05791-9
Published electronically: August 28, 2000
MathSciNet review: 1800240
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Abstract:

Extending results of a number of authors, we prove that if $U$ is the unipotent radical of an $\mathbb{R}$-split solvable epimorphic subgroup of a real algebraic group $G$ which is generated by unipotents, then the action of $U$ on $G/\Gamma$ is uniquely ergodic for every cocompact lattice $\Gamma$ in $G$. This gives examples of uniquely ergodic and minimal two-dimensional flows on homogeneous spaces of arbitrarily high dimension. Our main tools are the Ratner classification of ergodic invariant measures for the action of a unipotent subgroup on a homogeneous space, and a simple lemma (the `Cone Lemma') about representations of epimorphic subgroups.


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Additional Information

Barak Weiss
Affiliation: Department of Mathematics, State University of New York at Stony Brook, Stony Brook, New York 11794
Email: barak@math.sunysb.edu

DOI: https://doi.org/10.1090/S0002-9939-00-05791-9
Received by editor(s): April 22, 1999
Published electronically: August 28, 2000
Communicated by: Michael Handel
Article copyright: © Copyright 2000 American Mathematical Society

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