## Duke Mathematical Journal

### Distribution of periodic torus orbits on homogeneous spaces

#### Abstract

We prove results toward the equidistribution of certain families of periodic torus orbits on homogeneous spaces, with particular focus on the case of the diagonal torus acting on quotients of $\func{PGL}_n(\mathbb{R})$. After attaching to each periodic orbit an integral invariant (the discriminant), our results have the following flavor: certain standard conjectures about the distribution of such orbits hold up to exceptional sets of at most $O(\Delta^{\varepsilon})$ orbits of discriminant at most $\Delta$. The proof relies on the well-separatedness of periodic orbits together with measure rigidity for torus actions. We give examples of sequences of periodic orbits of this action that fail to become equidistributed even in higher rank. We also give an application of our results to sharpen a theorem of Minkowski on ideal classes in totally real number fields of cubic and higher degrees

#### Article information

Source
Duke Math. J., Volume 148, Number 1 (2009), 119-174.

Dates
First available in Project Euclid: 22 April 2009

https://projecteuclid.org/euclid.dmj/1240432194

Digital Object Identifier
doi:10.1215/00127094-2009-023

Mathematical Reviews number (MathSciNet)
MR2515103

Zentralblatt MATH identifier
1172.37003

#### Citation

Einsiedler, Manfred; Lindenstrauss, Elon; Michel, Philippe; Venkatesh, Akshay. Distribution of periodic torus orbits on homogeneous spaces. Duke Math. J. 148 (2009), no. 1, 119--174. doi:10.1215/00127094-2009-023. https://projecteuclid.org/euclid.dmj/1240432194

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