Ratner's theorems

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.

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.

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.

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. 

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.