Almost continuous orbit equivalence for non-singular homeomorphisms |
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Authors: | Alexandre I Danilenko Andrés del Junco |
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Institution: | (1) Department of Mathematics, California Institute of Technology 253-37, CA 91125 Pasadena, USA;(2) Department of Mathematics, University of California at Los Angeles, 6363 Math. Sci. Bldg, CA 90095-1555 Los Angeles, USA |
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Abstract: | Let X and Y be Polish spaces with non-atomic Borel measures μ and ν of full support. Suppose that T and S are ergodic non-singular homeomorphisms of (X, μ) and (Y, ν) with continuous Radon-Nikodym derivatives. Suppose that either they are both of type III
1 or that they are both of type III
λ, 0 < λ < 1 and, in the III
λ case, suppose in addition that both ‘topological asymptotic ranges’ (defined in the article) are log λ · ℤ. Then there exist invariant dense G
δ-subsets X′ ⊂ X and Y′ ⊂ Y of full measure and a non-singular homeomorphism ϕ: X′ → Y′ which is an orbit equivalence between T|
X′ and S|
Y′, that is ϕ{T
i
x} = {S
i
ϕx} for all x ∈ X′. Moreover, the Radon-Nikodym derivative dν ∘ ϕ/dμ is continuous on X′ and, letting S′ = ϕ
−1
Sϕ, we have T
x
= S′
n(x)
x and S′x = T
m(x)
x where n and m are continuous on X′. |
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Keywords: | |
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