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

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.

### MSC:

 28D05 Measure-preserving transformations

Zbl 0745.00063