Abstract
Ergodic one-parameter flows (G/Γ,gℝ) induced by the left action of a subgroupgℝ ⊂G on homogeneous spaces of finite volume are considered. Letℳ ⊂ ℝ+ be the set of allt>0 such that the cascade (G/Γ,gtℤ) is metrically isomorphic to the cascade (G/Γ,gℤ). We prove that eitherℳ is at most countable or the subgroupgℝ is horocyclic andℳ=ℝ+. We prove that a metric isomorphism of ergodic quasi-unipotent cascades (or flows) is affine on almost all fibers of a certain natural bundle. The result generalizes Witte's theorem on the affinity of such isomorphisms of cascades with the mixing property; this is applied to the study of the structure of the setℳ ⊂ ℝ+. The proof is based on the fundamental Ratner theorem stating that the ergodic measures of unipotent cascades are algebraic.
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