Hostname: page-component-8448b6f56d-dnltx Total loading time: 0 Render date: 2024-04-19T16:59:08.033Z Has data issue: false hasContentIssue false
  • English
  • Français

Equidistribution of divergent orbits of the diagonal group in the space of lattices

Part of: Ergodic theory

Published online by Cambridge University Press:  25 September 2018

OFIR DAVID Open the ORCID record for OFIR DAVID [Opens in a new window]  and
URI SHAPIRA
OFIR DAVID
Affiliation:
Department of Mathematics, Technion, Haifa, Israel email ushapira@tx.technion.ac.il
URI SHAPIRA
Affiliation:
Department of Mathematics, Hebrew University, Jerusalem, Israel email ofir.david@mail.huji.ac.il
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider divergent orbits of the group of diagonal matrices in the space of lattices in Euclidean space. We define two natural numerical invariants of such orbits: the discriminant—an integer—and the type—an integer vector. We then study the question of the limit distributional behavior of these orbits as the discriminant goes to infinity. Using entropy methods we prove that, for divergent orbits of a specific type, virtually any sequence of orbits equidistributes as the discriminant goes to infinity. Using measure rigidity for higher-rank diagonal actions, we complement this result and show that, in dimension three or higher, only very few of these divergent orbits can spend all of their life-span in a given compact set before they diverge.

Keywords

arithmetic and algebraic dynamicsnumber theorymaximal entropylocally finite measures

MSC classification

Primary: 37A17: Homogeneous flows
Secondary: 37A35: Entropy and other invariants, isomorphism, classification
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 5 , May 2020 , pp. 1217 - 1237
Copyright
© Cambridge University Press, 2018

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

David, O. and Shapira, U.. Equidistribution of divergent orbits and continued fraction expansion of rationals. J. Lond. Math. Soc. 98(1) (2018), 149–176.Google Scholar
Einsiedler, M., Katok, A. and Lindenstrauss, E.. Invariant measures and the set of exceptions to Littlewood’s conjecture. Ann. of Math. (2) 164(2) (2006), 513–560.Google Scholar
Einsiedler, M. and Lindenstrauss, E.. Diagonal actions on locally homogeneous spaces, in preparation.Google Scholar
Einsiedler, M., Lindenstrauss, E., Michel, P. and Venkatesh, A.. The distribution of closed geodesics on the modular surface, and Duke’s theorem. Enseign. Math. 58(3) (2012), 249–313.Google Scholar
Einsiedler, M., Lindenstrauss, E. and Thomas, W.. Entropy in ergodic theory and homogeneous dynamics, to appear. Preprint, available at http://www.personal.leeds.ac.uk/∼mattbw/entropy.Google Scholar
Shapira, U. and Zheng, C.. Limiting distributions of translates of divergent diagonal orbits. Preprint.Google Scholar
Tomanov, G. and Weiss, B.. Closed orbits for actions of maximal tori on homogeneous spaces. Duke Math. J. 119(2) (2003), 367–392.Google Scholar
Walter, P.. An Introduction to Ergodic Theory. Springer Science & Business Media, New York, 2000.Google Scholar