Stationary measures and invariant subsets of homogeneous spaces (III)

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Let G be a real Lie group, ? be a lattice in G and G be a compactly generated closed subgroup of G . If the Zariski closure of the group Ad(G) is semisimple with no compact factor, we prove that every G -orbit closure in G/? is a finite volume homogeneous space. We also establish related equidistribution properties.