Adic models of ergodic transformations, spectral theory, substitutions, and related topics. (English) Zbl 0770.28013

Vershik, A. M.; Livshits, A. N.

Representation theory and dynamical systems, Adv. Sov. Math. 9, 185-204 (1992).

[For the entire collection see Zbl 0745.00063.]
In this survey article the authors discuss the notion of an adic transformation as introduced by the first author. It comes from ideas about approximations in ergodic theory connected with the \(AF\)- and \(C^*\)-algebra approach to the study of skew products. Later, the second author noticed that stationary adic transformations are isomorphic to transformations given by substitutions. One of the main theorems of Vershik is that any ergodic transformation on a nonatomic Lebesgue space has an adic representation. The spectral properties are discussed. Adic representations of some measure-preserving transformations are given (rotation of the circle, rank 1 transformations, Chacon’s weakly mixing transformations, Ornstein-Shields non-Bernoulli \(K\)-automorphism and others). There are first examples of multidimensional adic transformations.

Reviewer: U.Krengel (Göttingen)

MSC:

28D05	Measure-preserving transformations
46L55	Noncommutative dynamical systems
46L05	General theory of \(C^*\)-algebras

Keywords:

substitutional transformation; adic transformation; ergodic transformation; spectral properties; measure-preserving transformations; rank 1 transformations; Chacon’s weakly mixing transformations; Ornstein- Shields non-Bernoulli \(K\)-automorphism; multidimensional adic transformations

Citations:

Zbl 0745.00063