Dynamical systems and ergodic theory Ergodic theory Theorem

Ratner's theorems

Builds on Lie group Lattice in a Lie group Probability measure Closure

Statement

Let $G$ be a connected Lie group, let $\Gamma$ be a lattice in $G$ , and let $X = G/\Gamma$ . An element $u \in G$ is unipotent if $\mathrm{Ad}(u)$ is a unipotent linear map of the Lie algebra of $G$ . Let $H$ be a connected closed subgroup of $G$ generated by one-parameter subgroups of unipotent elements. A probability measure $\mu$ on $X$ is homogeneous if there are a closed subgroup $L$ of $G$ and $x \in X$ with the following properties. The orbit $Lx$ is closed, and $\mu$ is the $L$ -invariant probability measure supported on $Lx$ . Then:

(Measure classification) every $H$ -invariant and ergodic Borel probability measure on $X$ is homogeneous;
(Orbit closures) for every $x \in X$ , the closure of $Hx$ equals $Lx$ for a closed subgroup $L \supseteq H$ such that $Lx$ carries an $L$ -invariant probability measure;
(Equidistribution) let $(u_t)_{t \in \R}$ be a one-parameter subgroup of unipotent elements, and let $x \in X$ . Then there is a homogeneous measure $\mu_x$ such that, for every bounded continuous $f$ on $X$ , $\lim_{T \to \infty} \frac{1}{T} \int_0^T f(u_t x) \, dt = \int_X f \, d\mu_x.$

Remarks

Marina Ratner proved these theorems around 1990 and 1991. The proof of the measure classification uses the polynomial divergence of nearby unipotent orbits: two such orbits drift apart slowly and mainly along the centraliser of the flow. Orbit closures and equidistribution then follow from the measure classification together with the non-divergence results of Margulis and Dani.

Example: $G = \mathrm{SL}_2(\R)$ , $\Gamma = \mathrm{SL}_2(\Z)$ , and $H = \set{\begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix} : t \in \R}$ . Then $X$ is the unit tangent bundle of the modular surface, and $H$ acts by the horocycle flow. Each $H$ -orbit is either a periodic closed horocycle or dense in $X$ , a fact known before Ratner; here $L = H$ or $L = G$ .

The unipotent hypothesis is necessary. For the diagonal subgroup of $\mathrm{SL}_2(\R)$ , acting by the geodesic flow, there are orbits whose closures are fractal sets, not orbits of any subgroup.

Going deeper

For a student

These theorems are central tools of homogeneous dynamics. Clause 2 is Raghunathan's conjecture. They have applications in number theory, such as the Oppenheim conjecture on values of indefinite quadratic forms, which Margulis had proved earlier by related methods. They also give counting results for integer points on homogeneous varieties.

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Closure concept
Lattice in a Lie group concept
Lie group concept
Probability measure concept
Ratner's theorems statement

Tag	`3MXR03096V`
URI	`m2://3MXR03096V`
Kind	statement, theorem
Writing	W0 stub
MSC	37A17, 22E40, 22D40
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About	Lattice in a Lie group
Uses	Lie group, Lattice in a Lie group, Probability measure, Closure
Same as	Wikidata Q7295875 Wikipedia Ratner's theorems
Changed	2 November 2026