# Ratner's theorems

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Ratner's theorems

In mathematics, Ratner's theorems is a group of major theorems in ergodic theory concerning unipotent flows on homogeneous spaces proved by Marina Ratner around 1990. The study of the dynamics of unipotent flows played decisive role in the proof of the Oppenheim conjecture by Margulis. Ratner's theorems and their later generalizations provide ways to both sharpen the results and extend the theory to the setting of arbitrary semisimple algebraic groups over a local field.

Short description

The Ratner orbit closure theorem asserts that the closures of orbits of unipotent flows on the quotient of a Lie group by a lattice are nice, geometric subsets. The Ratner equidistribution theorem further asserts that each such orbit is equidistributed in its closure. The Ratner measure classification theorem is the weaker statement that every ergodic invariant probability measure is homogeneous, or "algebraic": this turns out to be an important step towards proving the more general equidistribution property. There is no universal agreement on the names of these theorems: they are variously known as "measure rigidity theorem", "theorem on invariant measures" and its "topological version", and so on.

Let "G" be a Lie group, "Γ" a lattice in "G", and "u""t" a one-parameter subgroup of "G" consisting of unipotent elements, with the associated flow "φ""t" on "Γ""G". Then the closure of every orbit {"xu""t"} of "φ""t" is homogeneous. More precisely, there exists a connected, closed subgroup "S" of "G" such that the image of the orbit "xS" for the action of "S" by right translations on "G" under the canonical projection to "Γ""G" is closed, has a finite "S"-invariant measure, and contains the closure of the "φ""t"-orbit of "x" as a dense subset.

* Equidistribution theorem

References

Expositions

* Morris, Dave Witte, [http://people.uleth.ca/~dave.morris/lectures/Ratner/Ratner-v1-1.pdf "Ratner's Theorems on Unipotent Flows"] , Chicago Lectures in Mathematics, University of Chicago Press, 2005 ISBN 978-0-226-53984-3
* Marina Ratner, [http://www.scholarpedia.org/article/Ratner_theory "Ratner theory"] , Scholarpedia

Selected original articles

* M. Ratner, "Strict measure rigidity for unipotent subgroups of solvable groups", Invent. Math. 101 (1990), 449–482 MathSciNet|id=92h:22015
* M. Ratner, "On measure rigidity of unipotent subgroups of semisimple groups", Acta Math. 165 (1990), 229–309 MathSciNet|id=91m:57031
* M. Ratner, "On Raghunathan’s measure conjecture", Ann. of Math. 134 (1991), 545–607 MathSciNet|id=93a:22009
* M. Ratner, "Raghunathan’s topological conjecture and distributions of unipotent flows", Duke Math. J. 63 (1991), no. 1, 235–280 MathSciNet|id=93f:22012
* M. Ratner, " Raghunathan's conjectures for p-adic Lie groups", Internat. Math. Res. Notices ( 1993), 141-146.
* M. Ratner, " Raghunathan's conjectures for cartesian products of real and p-adic Lie groups ", Duke Math. J. 77(1995), no. 2, 275-382.
* G. A. Margulis and G. M. Tomanov, "Invariant measures for actions of unipotent groups over local fields on homogeneous spaces", Invent. Math. 116 (1994), 347–392 MathSciNet|id=95k:22013

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