- Ratner's theorems
In

mathematics ,**Ratner's theorems**is a group of major theorems inergodic theory concerning unipotent flows onhomogeneous space s proved byMarina Ratner around 1990. The study of the dynamics of unipotent flows played decisive role in the proof of theOppenheim 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 arbitrarysemisimple algebraic group s over alocal 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"}aone-parameter subgroup of "G" consisting ofunipotent 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 aconnected , 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 adense subset .**See also***

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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