Distribution of periodic torus orbits on homogeneous spaces. (English) Zbl 1172.37003
The authors obtain results toward the equidistribution of certain families of periodic torus orbits on homogeneous spaces. They focus on the case of diagonal torus actions on quotients of \(PGL_{n}({\mathbb R})\). By attaching to each periodic orbit an integral invariant (the discriminant), they obtain the result that certain standard conjectures about the distribution of such orbits hold up to exceptional sets of at most \(O(\Delta^{\epsilon})\) orbits of discriminant at most \(\Delta \). The proof relies on the facts that periodic orbits are well-separated and that torus actions are ‘measure-rigid’. Examples of sequences of periodic orbits of these actions that fail to become equidistributed (even in higher rank) are given. Also, the authors give an application of their results to sharpen a theorem of Minkowski on ideal classes in totally real number fields of cubic and higher degrees.
Reviewer: Bernd O. Stratmann (St. Andrews)
MSC:
| 37A17 | Homogeneous flows |
| 37A45 | Relations of ergodic theory with number theory and harmonic analysis (MSC2010) |
| 11E99 | Forms and linear algebraic groups |
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