Almost continuous orbit equivalence for non-singular homeomorphisms

作者：Danilenko Alexandre I^*; Del Junco Andres

来源：Israel Journal of Mathematics, 2011, 183(1): 165-188.

DOI：10.1007/s11856-011-0046-0

Let X and Y be Polish spaces with non-atomic Borel measures A mu and nu of full support. Suppose that T and S are ergodic non-singular homeomorphisms of (X, A mu) and (Y, nu) with continuous Radon-Nikodym derivatives. Suppose that either they are both of type III (1) or that they are both of type III (lambda), 0 < lambda < 1 and, in the III (lambda) case, suppose in addition that both 'topological asymptotic ranges' (defined in the article) are log lambda center dot a"currency sign. Then there exist invariant dense G (delta)-subsets X' aS, X and Y' aS, Y of full measure and a non-singular homeomorphism I center dot: X' -> Y' which is an orbit equivalence between T| (X') and S| (Y'), that is I center dot{T (i) x} = {S (i) I center dot x} for all x a X'. Moreover, the Radon-Nikodym derivative d nu a similar to I center dot/dA mu is continuous on X' and, letting S' = I center dot (-1) SI center dot, we have T (x) = S' (n(x)) x and S'x = T (m(x)) x where n and m are continuous on X'.

• 出版日期2011-6