Ratner's Theorems - Short Description

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 φ_t on Γ\G. Then the closure of every orbit {xut} 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.