On the spectral theory of adic transformations. (English) Zbl 0770.28012

Solomyak, B.

Representation theory and dynamical systems, Adv. Sov. Math. 9, 217-230 (1992).

[For the entire collection see Zbl 0745.00063.]
The spectral properties of stationary adic transformations are studied. Sufficient conditions for existence and nonexistence of nonconstant eigenvectors are given. A sufficient condition for a purely discrete spectrum is applied to a series of examples. Using the results of Livshits on equivalence of substitutional flows and adic transformations the main Theorem 4.1 asserts: The minimal flow generated by the substitution on \(\{1,\dots,m\}\) with \(1\to 123\cdots m\), \(i\to i-1\) \((i>1)\), is metrically isomorphic to a translation on the \((m-1)\)- dimensional torus.

Reviewer: U.Krengel (Göttingen)

MSC:

28D05

Measure-preserving transformations

Keywords:

spectral properties; stationary adic transformations; substitutional flows; minimal flow

Citations:

Zbl 0745.00063