Isomorphic extensions and applications. (English) Zbl 1362.37026

Downarowicz, Tomasz; Glasner, Eli

Topol. Methods Nonlinear Anal. 48, No. 1, 321-338 (2016).

Summary: If \(\pi\colon (X,T)\to(Z,S)\) is a topological factor map between uniquely ergodic topological dynamical systems, then \((X,T)\) is called an isomorphic extension of \((Z,S)\) if \(\pi\) is also a measure-theoretic isomorphism. We consider the case when the systems are minimal and we pay special attention to equicontinuous \((Z,S)\). We first establish a characterization of this type of isomorphic extensions in terms of mean equicontinuity, and then show that an isomorphic extension need not be almost one-to-one, answering questions of J. Li et al. [Ergodic Theory Dyn. Syst. 35, No. 8, 2587–2612 (2015; Zbl 1356.37016)].

MSC:

37B05	Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37A05	Dynamical aspects of measure-preserving transformations

Keywords:

minimality; unique ergodicity; isomorphic extension; almost one-to-one extension; mean equicontinuity; skew product

Citations:

Zbl 1356.37016