Mesures stationnaires et fermés invariants des espaces homogènes

Abstract

Let G be a real simple Lie group, Λ be a lattice of G and Γ be a Zariski dense subsemigroup of G. We prove that every infinite Γ-invariant subset in the quotient X = G/Λ is dense. Let μ be a probability measure on G whose support is compact and spans a Zariski dense subgroup of G. We prove that every atom free μ-stationary probability measure on X is G-invariant. We also prove similar results for the torus X = Td.