# Ratner's theorems

Ratner's theorems This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. Please help to improve this article by introducing more precise citations. (September 2019) (Learn how and when to remove this template message) In mathematics, Ratner's theorems are a group of major theorems in ergodic theory concerning unipotent flows on homogeneous spaces proved by Marina Ratner around 1990. The theorems grew out of Ratner's earlier work on horocycle flows. The study of the dynamics of unipotent flows played a decisive role in the proof of the Oppenheim conjecture by Grigory Margulis. Ratner's theorems have guided key advances in the understanding of the dynamics of unipotent flows. 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.

Contents 1 Short description 1.1 Example: '"UNIQ--postMath-00000013-QINU"' 2 See also 3 References 3.1 Expositions 3.2 Selected original articles 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 the "measure rigidity theorem", the "theorem on invariant measures" and its "topological version", and so on.

The formal statement of such a result is as follows. Let {displaystyle G} be a Lie group, {displaystyle {mathit {Gamma }}} a lattice in {displaystyle G} , and {displaystyle u^{t}} a one-parameter subgroup of {displaystyle G} consisting of unipotent elements, with the associated flow {displaystyle phi _{t}} on {displaystyle {mathit {Gamma }}setminus G} . Then the closure of every orbit {displaystyle left{xu^{t}right}} of {displaystyle phi _{t}} is homogeneous. This means that there exists a connected, closed subgroup {displaystyle S} of {displaystyle G} such that the image of the orbit {displaystyle ,xS,} for the action of {displaystyle S} by right translations on {displaystyle G} under the canonical projection to {displaystyle {mathit {Gamma }}setminus G} is closed, has a finite {displaystyle S} -invariant measure, and contains the closure of the {displaystyle phi _{t}} -orbit of {displaystyle x} as a dense subset.

Example: {displaystyle SL_{2}(mathbb {R} )} The simplest case to which the statement above applies is {displaystyle G=SL_{2}(mathbb {R} )} . In this case it takes the following more explicit form; let {displaystyle Gamma } be a lattice in {displaystyle SL_{2}(mathbb {R} )} and {displaystyle Fsubset Gamma backslash G} a closed subset which is invariant under all maps {displaystyle Gamma gmapsto Gamma (gu_{t})} where {displaystyle u_{t}={begin{pmatrix}1&t\0&1end{pmatrix}}} . Then either there exists an {displaystyle xin Gamma backslash G} such that {displaystyle F=xU} (where {displaystyle U={u_{t},tin mathbb {R} }} ) or {displaystyle F=Gamma backslash G} .

In geometric terms {displaystyle Gamma } is a cofinite Fuchsian group, so the quotient {displaystyle M=Gamma backslash mathbb {H} ^{2}} of the hyperbolic plane by {displaystyle Gamma } is a hyperbolic orbifold of finite volume. The theorem above implies that every horocycle of {displaystyle mathbb {H} ^{2}} has an image in {displaystyle M} which is either a closed curve (a horocycle around a cusp of {displaystyle M} ) or dense in {displaystyle M} .

See also Equidistribution theorem References Expositions Morris, Dave Witte (2005). Ratner's Theorems on Unipotent Flows (PDF). Chicago Lectures in Mathematics. Chicago, IL: University of Chicago Press. ISBN 978-0-226-53984-3. MR 2158954. Einsiedler, Manfred (2009). "What is... measure rigidity?" (PDF). Notices of the AMS. 56 (5): 600–601. Selected original articles Ratner, Marina (1990). "Strict measure rigidity for unipotent subgroups of solvable groups". Invent. Math. 101 (2): 449–482. doi:10.1007/BF01231511. MR 1062971. Ratner, Marina (1990). "On measure rigidity of unipotent subgroups of semisimple groups". Acta Math. 165 (1): 229–309. doi:10.1007/BF02391906. MR 1075042. Ratner, Marina (1991). "On Raghunathan's measure conjecture". Ann. of Math. 134 (3): 545–607. doi:10.2307/2944357. MR 1135878. Ratner, Marina (1991). "Raghunathan's topological conjecture and distributions of unipotent flows". Duke Math. J. 63 (1): 235–280. doi:10.1215/S0012-7094-91-06311-8. MR 1106945. Ratner, Marina (1993). "Raghunathan's conjectures for p-adic Lie groups". International Mathematics Research Notices (5): 141–146. doi:10.1155/S1073792893000145. MR 1219864. Ratner, Marina (1995). "Raghunathan's conjectures for cartesian products of real and p-adic Lie groups". Duke Math. J. 77 (2): 275–382. doi:10.1215/S0012-7094-95-07710-2. MR 1321062. Margulis, Grigory A.; Tomanov, Georges M. (1994). "Invariant measures for actions of unipotent groups over local fields on homogeneous spaces". Invent. Math. 116 (1): 347–392. doi:10.1007/BF01231565. MR 1253197. Categories: Ergodic theoryLie groupsTheorems in dynamical systems

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