---
tag: 3MXR03096V
kind: statement
role: theorem
name: Ratner's theorems
msc: [37A17, "22E40", 22D40]
uses: [7JBRGCVJRB, BTYF5Y5PFY, 059TJN0NZ1, 0QEAJB91GZ]
about: [BTYF5Y5PFY]
sources:
  - url: https://en.wikipedia.org/wiki/Ratner's_theorems
same_as:
  wikidata: Q7295875
  wikipedia: Ratner's theorems
---

## Statement

Let $G$ be a connected [[Lie group|7JBRGCVJRB]], let $\Gamma$ be a [[lattice|BTYF5Y5PFY]] 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|059TJN0NZ1]] supported on $Lx$.
Then:

1. (Measure classification) every $H$-invariant and ergodic Borel probability measure on $X$ is homogeneous;
2. (Orbit closures) for every $x \in X$, the [[closure|0QEAJB91GZ]] of $Hx$ equals $Lx$ for a closed subgroup $L \supseteq H$ such that $Lx$ carries an $L$-invariant probability measure;
3. (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.

## Explanations

### L1 human

Imagine a flow on a finite-volume space built from a Lie group, such as the motion of horocycles on a hyperbolic surface.
For unipotent flows, Ratner proved that the behaviour is always very orderly.
Each orbit either closes up or fills out a nice smooth piece of the space evenly.
There are no strange fractal-shaped orbit closures, which do occur for other flows like the geodesic flow.

### L2 human

These theorems are central tools of homogeneous dynamics.
Clause 2 is [[Raghunathan's conjecture|ME4ZSRAT5Z]].
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.