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 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.
Read more about this topic: Ratner's Theorems
Famous quotes containing the words short and/or description:
“Man that is born of woman hath but a short time to live, and is full of misery. He cometh up, and is cut down, like a flower; he fleeth as it were a shadow, and never continueth in one stay.”
—Book Of Common Prayer, The. Burial of the Dead, “First Anthem,” (1662)
“An intentional object is given by a word or a phrase which gives a description under which.”
—Gertrude Elizabeth Margaret Anscombe (b. 1919)