Lacunary isomorphism of decreasing sequences of measurable partitions (Q1355270)

Let \((X,{\mathcal F},m)\) be a Lebesgue space with \(m(X)=1\), and let \(\xi=\{\xi_n\}^\infty_{n=0}\) be a decreasing sequence of measurable partitions (d.s.m.p), where \(\xi_0=\varepsilon\) denotes the partition of \(X\) into separate points. Two d.s.m.p. \(\xi= \{\xi_n\}\) and \(\xi'=\{\xi_n'\}\) are called (i) isomorphic (notation: \(\xi\overset {I}\sim\xi'\)), if \(\Phi(\xi_n)=\xi_n'\), \(n\geq 1\), where \(\Phi:X\to X'\) is an isomorphism; (ii) lacunary isomorphic (\(\xi\overset{LI}\sim \xi'\)), if there exist \(n_1<n_2<n_3<\cdots\) such that subsequences \(\{\xi_n\}\) and \(\{\xi_n'\}\) are isomorphic; (iii) finitely isomorphic \((\xi\overset{FI}\sim \xi'\)), if for any \(n\) there exists an isomorphisms \(\Phi_n: X\to X'\) such that \(\Phi_n(\xi_k)= \xi_k'\), \(k=1,\dots,n\); (iv) orbitally isomorphic (\(\xi\overset{OI}\sim \xi'\)), if the intersections \(\theta(\xi)=\bigcap^\infty_{n=1}\xi_n\) and \(\theta(\xi')= \bigcap^\infty_{n=1}\xi_n'\) are isomorphic, where the partition \(\theta(\xi)\) (not necessary measurable) is defined by \(x\overset{\theta(\xi)} \sim y\Leftrightarrow\exists n:x\overset{\xi_n}\sim y\), \((x,y)\in X\times X'\). It is obvious that \(FI\Leftarrow I\Rightarrow LI\Rightarrow OI\). A sequence \(\xi\) is called ergodic if the measurable intersection \(\bigwedge^\infty_{n=0}\xi_n\) (i.e., the measurable hull of \(\theta(\xi)\)) is trivial. The author considers in this paper two problems: A. When for ergodic d.s.m.p. does \(FI+OI\Rightarrow LI\)? B. When are ergodic finitely isomorphic sequences orbitally isomorphic? The solutions of both of these problems are described for a wide class of finite Bernoulli sequences of measurable partitions.