Some applications of groups of essential values of cocycles in topological dynamics (Q1772557)

In the first part of the paper (Section 3) some examples are presented which show that not all statements known for measure theoretic extensions have their counterparts in the topological setting (e.g. a cocycle \(\phi: X\to G\), \(G\) an Abelian group, is regular iff the group of essential values \(E_\infty(\widetilde\phi)\) is trivial, where \(\widetilde\phi: X\to G/E(\phi)\) is defined by \(\widetilde\phi(x)= \phi(x)E\phi)\) [cf. \textit{K. Schmidt}, Cocycles on ergodic transformation groups (MacMillan, Dehli) (1977; Zbl 0421.28017)]). In Section 4 the author investigates base preserving equivariant homeomorphisms of Rokhlin cocycle extensions of minimal flows, obtaining results in the topological setting and which are motivated by work of \textit{M. Lemanczyk} and \textit{E. Lesigne} [J. Anal. Math. 85, 43--86 (2001; Zbl 1038.37003)] and \textit{M. Lemanczyk} and \textit{F. Parreau} [Ergodic Theory Dyn. Syst. 23, No. 5, 1525--1550 (2003; Zbl 1043.37004)] in the measure theoretic setting.