Minimality and unique ergodicity for subgroup actions

Shahar Mozes, Barak Weiss

Type: Article

Publication Date: 1998-01-01

Citations: 7

DOI: https://doi.org/10.5802/aif.1665

Abstract

Let G be an ℝ-algebraic semisimple group, H an algebraic ℝ-subgroup, and Γ a lattice in G. Partially answering a question posed by Hillel Furstenberg in 1972, we prove that if the action of H on G/Γ is minimal, then it is uniquely ergodic. Our proof uses in an essential way Marina Ratner’s classification of probability measures on G/Γ invariant under unipotent elements, and the study of “tubes” in G/Γ.

Locations

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Works Cited by This (8)

Action	Title	Year	Authors
+	Limit distributions of orbits of unipotent flows and values of quadratic forms	1993	S. G. Dani G. A. Margulis
+	Finite dimensional representations and subgroup actions on homogeneous spaces	1998	Barak Weiss
+	Cartan Subgroups and Lattices in Semi-Simple Groups	1972	Gopal Prasad M. S. Raghunathan
+	Measure rigidity for almost linear groups and its applications	1996	G. A. Margulis Georges Tomanov
+	Invariant measures and orbit closures for unipotent actions on homogeneous spaces	1994	Marina Ratner
+	On ergodic quasi-invariant measures of group automorphism	1982	S. G. Dani
+	Minimal sets of homogeneous flows	1995	A. N. Starkov
+	Epimorphic subgroups and invariant measures	1995	Shahar Mozes