Factor orbit equivalence of compact group extensions and classification of finite extensions of ergodic automorphisms (Q580529)

Let S and \(S'\) be ergodic measure preserving automorphisms of the Lebesgue spaces \((Y,{\mathcal C},\nu)\) and \((Y',{\mathcal C}',\nu '),\) respectively. H. Dye proved in 1958 that such S and \(S'\) are orbit equivalent. Suppose that \({\mathcal B}\) is an S-invariant \(\sigma\)-subfield of \({\mathcal C}\) such that the quotient is a Lebesgue space again and make the same hypothesis on \({\mathcal B}'\subset {\mathcal C}'.\) The author investigates the existence of an isomorphism \(\Psi:(Y,C,\nu)\to (Y',C',\nu ')\) which on the one hand establishes an orbit equivalence and on the other hand sends \({\mathcal B}\) into \({\mathcal B}'\). The techniques extend to give another proof of a result due to \textit{A. Fieldsteel} [Isr. J. Math. 38, 289-303 (1981; Zbl 0461.28013)] on factor orbit equivalence of compact group extensions.