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1 results

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

  • Part of
  • OFIR DAVID, URI SHAPIRA
  • Journal:
    Ergodic Theory and Dynamical Systems / Volume 40 / Issue 5 / May 2020
    Published online by Cambridge University Press:
    25 September 2018, pp. 1217-1237
    Print publication:
    May 2020

  • 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.