Systems of finite rank. (English) Zbl 0883.28014

Ferenczi, Sébastien

Colloq. Math. 73, No. 1, 35-65 (1997).

This paper contains a survey of a special class of examples of systems constructed in ergodic theory. Its aim is to gather definitions related to notions of rank, to describe the relations between them, and to show how these notions apply to some classical systems. At the beginning, some fundamental notions are given: measurable dynamical systems, measurable isomorphisms, ergodicity, mixing, rigidity, partitions, entropy. Then the author discusses systems of rank one. He presents a nonconstructive and constructive geometric definition, and nonconstructive/constructive symbolic one. Examples of systems defined according to these definitions are given, and some “famous” rank systems are described. The author also gives some measure theoretic properties of rank one systems. Finally, systems of rank at most \(r\) are defined and their properties enumerated. The last part of the paper is concerned with rank properties of classical systems.

Reviewer: E.Petrisor (Timişoara)

MSC:

28D10

One-parameter continuous families of measure-preserving transformations

Keywords:

measure preserving systems; ergodicity; entropy; systems of rank one