Measure-theoretic problems in topological dynamics
In this interesting paper, the author is concerned with point realizations of \(\sigma\)-homomorphisms, with compact topological transformation groups, and with distal flows. In the principal Theorem 3.7, it is shown that the phase space \(X\) of a minimal distal flow \((X,T)\) is Baire isomorphic to a product of compact metric spaces, and the isomorphism takes some invariant completion-regular probability measure to a direct product measure. This theorem is a significant advance in the literature, and has significant consequences. For instance: If \(\nu\) is any finite Baire measure on \(X\) (as above), then every automorphisms of the measure algebra of \(\nu\) is induced by an invertible completion Baire point transformation of \(X\). If \(\lambda\), \(\nu\) are Radon probability measures on \(X\) as above, and if \(\exists\) a measure-preserving isomorphism of their measure algebras, then \(\lambda\) and \(\nu\) are completion Baire isomorphic. Correction: The author of reference [23] in the bibliography (p. 220) is D. McMahon.
compact topological transformation groups
invertible completion Baire point transformation
scientific article; zbMATH DE number 807407
scientific article