Abstract
Suppose X and Y are Polish spaces with non-atomic Borel probability measures μ and ν and suppose that T and S are ergodic measure-preserving homeomorphisms of (X, μ) and (Y, ν). Then there are invariant G δ subsets X′ ⊂ X and Y′ ⊂ Y of full measure and a homeomorphism ϕ: X′ → Y′ which maps μ|X′ to ν|Y′ and maps T-orbits onto S-orbits. We also deal with the case where T and S preserve infinite invariant measures.
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del Junco, A., Şahin, A. Dye’s theorem in the almost continuous category. Isr. J. Math. 173, 235–251 (2009). https://doi.org/10.1007/s11856-009-0090-1
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DOI: https://doi.org/10.1007/s11856-009-0090-1