TY - JOUR

T1 - On sets invariant under the action of the diagonal group

AU - Lindenstrauss, Elon

AU - Weiss, Barak

PY - 2001/10/1

Y1 - 2001/10/1

N2 - We consider the action of the (n - 1)-dimensional group of diagonal matrices in SL(n, ℝ) on SL(n, ℝ)/Γ, where Γ is a lattice and n ≥ 3. Far-reaching conjectures of Furstenberg, Katok-Spatzier and Margulis suggest that there are very few closed invariant sets for this action. We examine the closed invariant sets containing compact orbits. For example, for Γ = SL(n, ℤ) we describe all possible orbit-closures containing a compact orbit. In marked contrast to the case n = 2, such orbit-closures are necessarily homogeneous submanifolds in the sense of Ratner.

AB - We consider the action of the (n - 1)-dimensional group of diagonal matrices in SL(n, ℝ) on SL(n, ℝ)/Γ, where Γ is a lattice and n ≥ 3. Far-reaching conjectures of Furstenberg, Katok-Spatzier and Margulis suggest that there are very few closed invariant sets for this action. We examine the closed invariant sets containing compact orbits. For example, for Γ = SL(n, ℤ) we describe all possible orbit-closures containing a compact orbit. In marked contrast to the case n = 2, such orbit-closures are necessarily homogeneous submanifolds in the sense of Ratner.

UR - http://www.scopus.com/inward/record.url?scp=0035623098&partnerID=8YFLogxK

U2 - 10.1017/S0143385701001717

DO - 10.1017/S0143385701001717

M3 - Article

AN - SCOPUS:0035623098

VL - 21

SP - 1481

EP - 1500

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

SN - 0143-3857

IS - 5

ER -