## Abstract

For an inclusion F < G < L of connected real algebraic groups such that F is epimorphic in G, we show that any closed F-invariant subset of L/Λ is G-invariant, where Λ is a lattice in L. This is a topological analogue of a result due to S. Mozes, that any finite F-invariant measure on L/Λ is G-invariant. This result is established by proving the following result. If in addition G is generated by unipotent elements, then there exists a ∈ F such that the following holds. Let U ⊂ F be the subgroup generated by all unipotent elements of F, x ∈ L/Λ, and λ and μ denote the Haar probability measures on the homogeneous spaces Ux and Gx, respectively (cf. Ratner's theorem). Then a^{n}λ → μ weakly as n → ∞. We also give an algebraic characterization of algebraic subgroups F < SL_{n}(ℝ) for which all orbit closures on SL_{n}(ℝ)/SL_{n}(ℤ) are finite-volume almost homogeneous, namely the smallest observable subgroup of SL_{n}(ℝ) containing F should have no non-trivial algebraic characters defined over ℝ.

Original language | English |
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Pages (from-to) | 567-592 |

Number of pages | 26 |

Journal | Ergodic Theory and Dynamical Systems |

Volume | 20 |

Issue number | 2 |

DOIs | |

State | Published - 1 Jan 2000 |

Externally published | Yes |

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics