Ratner's theorems
Encyclopedia
In mathematics
, Ratner's theorems are a group of major theorems in ergodic theory
concerning unipotent flows on homogeneous space
s 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 group
s over a local field
.
Let G be a Lie group
, Γ a lattice
in G, and ut a one-parameter subgroup of G consisting of unipotent
elements, with the associated flow
φt on Γ\G. Then the closure of every orbit {xut} of φt 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 φt-orbit of x as a dense subset.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Ratner's theorems are a group of major theorems in ergodic theory
Ergodic theory
Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
concerning unipotent flows on homogeneous space
Homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...
s proved by Marina Ratner
Marina Ratner
Marina Ratner is a professor of mathematics at the University of California, Berkeley who works in ergodic theory. Around 1990 she proved a group of major theorems concerning unipotent flows on homogeneous spaces, known as Ratner's theorems. Ratner was awarded the Ostrowski Prize in 1993 and...
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
Oppenheim conjecture
In Diophantine approximation, the Oppenheim conjecture concerns representations of numbers by real quadratic forms in several variables. It was formulated in 1929 by Alexander Oppenheim and later the conjectured property was further strengthened by Davenport and Oppenheim...
by Grigory Margulis
Grigory Margulis
Gregori Aleksandrovich Margulis is a Russian mathematician known for his far-reaching work on lattices in Lie groups, and the introduction of methods from ergodic theory into diophantine approximation. He was awarded a Fields Medal in 1978 and a Wolf Prize in Mathematics in 2005, becoming the...
. 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 group
Semisimple algebraic group
In mathematics, especially in the areas of abstract algebra and algebraic geometry studying linear algebraic groups, a semisimple algebraic group is a type of matrix group which behaves much like a semisimple Lie algebra or semisimple ring.- Definition :...
s over a local field
Local field
In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...
.
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
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
, Γ a lattice
Lattice (discrete subgroup)
In Lie theory and related areas of mathematics, a lattice in a locally compact topological group is a discrete subgroup with the property that the quotient space has finite invariant measure...
in G, and ut a one-parameter subgroup of G consisting of unipotent
Unipotent
In mathematics, a unipotent element r of a ring R is one such that r − 1 is a nilpotent element, in other words such that some power n is zero....
elements, with the associated flow
Flow (mathematics)
In mathematics, a flow formalizes the idea of the motion of particles in a fluid. Flows are ubiquitous in science, including engineering and physics. The notion of flow is basic to the study of ordinary differential equations. Informally, a flow may be viewed as a continuous motion of points over...
φt on Γ\G. Then the closure of every orbit {xut} of φt is homogeneous. More precisely, there exists a connected
Connectedness
In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is connected; otherwise it is disconnected...
, 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 φt-orbit of x as a dense subset.
Expositions
- Morris, Dave Witte, Ratner's Theorems on Unipotent Flows, Chicago Lectures in Mathematics, University of Chicago Press, 2005 ISBN 978-0-226-53984-3
- Marina RatnerMarina RatnerMarina Ratner is a professor of mathematics at the University of California, Berkeley who works in ergodic theory. Around 1990 she proved a group of major theorems concerning unipotent flows on homogeneous spaces, known as Ratner's theorems. Ratner was awarded the Ostrowski Prize in 1993 and...
, Ratner theory, ScholarpediaScholarpediaScholarpedia is an English-language online wiki-based encyclopedia that uses the same MediaWiki software as Wikipedia, but has features more commonly associated with open-access online academic journals....
Selected original articles
- M. RatnerMarina RatnerMarina Ratner is a professor of mathematics at the University of California, Berkeley who works in ergodic theory. Around 1990 she proved a group of major theorems concerning unipotent flows on homogeneous spaces, known as Ratner's theorems. Ratner was awarded the Ostrowski Prize in 1993 and...
, Strict measure rigidity for unipotent subgroups of solvable groups, Invent. Math. 101 (1990), 449–482 - M. Ratner, On measure rigidity of unipotent subgroups of semisimple groups, Acta Math. 165 (1990), 229–309
- M. Ratner, On Raghunathan’s measure conjecture, Ann. of Math. 134 (1991), 545–607
- M. Ratner, Raghunathan’s topological conjecture and distributions of unipotent flows, Duke Math. J. 63 (1991), no. 1, 235–280
- 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. MargulisGrigory MargulisGregori Aleksandrovich Margulis is a Russian mathematician known for his far-reaching work on lattices in Lie groups, and the introduction of methods from ergodic theory into diophantine approximation. He was awarded a Fields Medal in 1978 and a Wolf Prize in Mathematics in 2005, becoming the...
and G. M. Tomanov, Invariant measures for actions of unipotent groups over local fields on homogeneous spaces, Invent. Math. 116 (1994), 347–392