Ratner's theorems are a collection of fundamental results in ergodic theory and homogeneous dynamics, proved by the mathematician Marina Ratner in the early 1990s. The theorems describe the behavior of actions of unipotent subgroups on homogeneous spaces of Lie groups, providing precise classifications of invariant measures, orbit closures, and equidistribution properties.
Key Results
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Measure Classification Theorem
For a Lie group $G$ and a lattice $\Gamma \subset G$, let $U$ be a subgroup generated by unipotent elements. Any $U$-invariant and ergodic probability measure on the homogeneous space $G/\Gamma$ is algebraic: it is the homogeneous (Haar) measure on a closed orbit of a subgroup $H$ containing $U$. In particular, such measures are supported on orbits of intermediate subgroups $H$ with $U \subset H \subset G$. -
Orbit Closure Theorem
The closure of any orbit of a unipotent flow $U$ acting on $G/\Gamma$ is also homogeneous: it is the orbit of a closed subgroup $H$ containing $U$. Consequently, each orbit is uniformly distributed with respect to the Haar measure on its closure. -
Equidistribution Theorem
Sequences of suitably normalized pieces of unipotent orbits become equidistributed in their closures, converging to the Haar measure on the corresponding homogeneous subspace.
Historical Context
- The study of unipotent flows dates to earlier work by Hedlund (1932) on geodesic flows on the modular surface and by Margulis (1970s) on the Oppenheim conjecture.
- Ratner’s theorems extended these earlier partial results to a broad, systematic framework applicable to arbitrary Lie groups and lattices.
- The proofs combined techniques from ergodic theory, algebraic groups, and Lie theory, introducing new methods such as “Ratner’s joinings” and “measure rigidity”.
Applications
- Number Theory: Proof of the Oppenheim conjecture on values of indefinite quadratic forms (originally achieved by Margulis; Ratner’s results provide alternative approaches).
- Arithmetic Geometry: Distribution of rational points on homogeneous varieties.
- Quantum Chaos: Results on quantum unique ergodicity for arithmetic manifolds.
- Rigidity Theory: Foundations for subsequent rigidity theorems, including those of Eskin–Mirzakhani–Mohammadi on moduli spaces of translation surfaces.
References
- Ratner, Marina. “On measure rigidity of unipotent subgroups of semisimple groups.” Annals of Mathematics 134 (1991): 245–271.
- Ratner, Marina. “Raghunathan’s conjectures for Cartan actions on homogeneous spaces.” Israel Journal of Mathematics 86 (1994): 319–345.
- Einsiedler, Manfred; Katok, Anatole; Lindenstrauss, Elon. Invariant measures and the set of exceptions to Littlewood’s conjecture. Annals of Mathematics Studies, Princeton University Press, 2011.
- Einsiedler, Manfred; Ward, Thomas. Ergodic Theory with a View Toward Number Theory. Springer, 2011 (Chapter 8 covers Ratner’s theorems).
These results remain central to modern research in dynamics, geometry, and number theory.
