Ratner's theorems explained
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.
The formal statement of such a result is as follows. Let
be a
Lie group,
a
lattice in
, and
a
one-parameter subgroup of
consisting of
unipotent elements, with the associated
flow
on
. Then the closure of every orbit
of
is homogeneous. This means that there exists a
connected, closed subgroup
of
such that the image of the orbit
for the action of
by right translations on
under the canonical projection to
is closed, has a finite
-invariant measure, and contains the closure of the
-orbit of
as a
dense subset.
Example:
The simplest case to which the statement above applies is
. In this case it takes the following more explicit form; let
be a lattice in
and
F\subset\Gamma\backslashG
a closed subset which is invariant under all maps
\Gammag\mapsto\Gamma(gut)
where
ut=\begin{pmatrix}1&t\ 0&1\end{pmatrix}
. Then either there exists an
such that
(where
) or
.
In geometric terms
is a cofinite
Fuchsian group, so the quotient
of the
hyperbolic plane by
is a hyperbolic
orbifold of finite volume. The theorem above implies that every
horocycle of
has an image in
which is either a closed curve (a horocycle around a
cusp of
) or dense in
.
See also
References
Expositions
Selected original articles
- Marina . Ratner . Strict measure rigidity for unipotent subgroups of solvable groups . . 101 . 2 . 1990 . 449–482 . 1062971 . 10.1007/BF01231511.
- Marina . Ratner . On measure rigidity of unipotent subgroups of semisimple groups . . 165 . 1990 . 229–309 . 1075042 . 10.1007/BF02391906 . 1. free .
- Marina . Ratner . On Raghunathan's measure conjecture . . 134 . 3 . 1991 . 545–607 . 1135878 . 10.2307/2944357. 2944357 .
- Marina . Ratner . Raghunathan's topological conjecture and distributions of unipotent flows . . 63 . 1991 . 1 . 235–280 . 10.1215/S0012-7094-91-06311-8 . 1106945.
- Marina . Ratner . Raghunathan's conjectures for p-adic Lie groups . . 1993 . 1993 . 5 . 141–146 . 10.1155/S1073792893000145 . 1219864 . free .
- Marina . Ratner . Raghunathan's conjectures for cartesian products of real and p-adic Lie groups . . 77 . 1995 . 2 . 275–382 . 10.1215/S0012-7094-95-07710-2 . 1321062.
- Grigory Margulis . Grigory A. . Margulis . Georges M. . Tomanov . Invariant measures for actions of unipotent groups over local fields on homogeneous spaces . . 116 . 1994 . 347–392 . 1253197 . 1 . 10.1007/BF01231565.
