Ratner's theorems

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 ${\displaystyle G}$ be a Lie group, ${\displaystyle \mathit{\Gamma }}$ a lattice in ${\displaystyle G}$, and ${\displaystyle u^t}$ a one-parameter subgroup of ${\displaystyle G}$ consisting of unipotent elements, with the associated flow ${\displaystyle {\varphi }_t}$ on ${\displaystyle \mathit{\Gamma }\setminus G}$. Then the closure of every orbit ${\displaystyle \left\{x{u}^t\right\}}$ of ${\displaystyle {\varphi }_t}$ is homogeneous. This means that there exists a connected, closed subgroup ${\displaystyle S}$ of ${\displaystyle G}$ such that the image of the orbit ${\displaystyle xS}$ for the action of ${\displaystyle S}$ by right translations on ${\displaystyle G}$ under the canonical projection to ${\displaystyle \mathit{\Gamma }\setminus G}$ is closed, has a finite ${\displaystyle S}$-invariant measure, and contains the closure of the ${\displaystyle {\varphi }_t}$-orbit of ${\displaystyle x}$ as a dense subset.

Example: S L 2 ( R ) {\displaystyle SL_{2}(\mathbb {R} )}

The simplest case to which the statement above applies is ${\displaystyle G=S{L}_2(\mathbb{R})}$. In this case it takes the following more explicit form; let ${\displaystyle \mathrm{\Gamma }}$ be a lattice in ${\displaystyle S{L}_2(\mathbb{R})}$ and ${\displaystyle F\subset \mathrm{\Gamma }\mathrm{\setminus }G}$ a closed subset which is invariant under all maps ${\displaystyle \mathrm{\Gamma }g\mapsto \mathrm{\Gamma }(g{u}_t)}$ where ${\displaystyle u_t=\left(\begin{array}{cc}1& t\\ 0& 1\end{array}\right)}$. Then either there exists an ${\displaystyle x\in \mathrm{\Gamma }\mathrm{\setminus }G}$ such that ${\displaystyle F=xU}$ (where ${\displaystyle U=\{u_t,t\in \mathbb{R}\}}$) or ${\displaystyle F=\mathrm{\Gamma }\mathrm{\setminus }G}$.

In geometric terms ${\displaystyle \mathrm{\Gamma }}$ is a cofinite Fuchsian group, so the quotient ${\displaystyle M=\mathrm{\Gamma }\mathrm{\setminus }{\mathbb{H}}^2}$ of the hyperbolic plane by ${\displaystyle \mathrm{\Gamma }}$ is a hyperbolic orbifold of finite volume. The theorem above implies that every horocycle of ${\displaystyle {\mathbb{H}}^2}$ has an image in ${\displaystyle M}$ which is either a closed curve (a horocycle around a cusp of ${\displaystyle M}$) or dense in ${\displaystyle M}$.