On the multiplicity function of ergodic group extensions of rotations. (English) Zbl 0830.28009

Goodson, G. R.; Kwiatkowski, J.; Lemańczyk, M.; Liardet, P.

Stud. Math. 102, No. 2, 157-174 (1992).

Let \({\mathcal M}_T\subset \mathbb{N}\cup \{\infty\}\) be the set of all essential spectral multiplicities of an ergodic measure-preserving transformation \(T\) of a Lebesgue space. The authors address the problem of finding all sets \(A\) which can serve as \({\mathcal M}_T\) for an appropriate \(T\). E. A. Robinson jun. [Lect. Notes Math. 1342, 645- 652 (1988; Zbl 0678.28009)] showed the existence of such \(T\) for all finite sets \(A\subset \mathbb{N}\) satisfying the following conditions: \(1\in A\), and \(\text{lcm}(m_1, m_2)\in A\) whenever \(m_1, m_2\in A\).
The authors prove that an ergodic transformation \(T\) with \({\mathcal M}_T= A\) exists for any \(A\subset \mathbb{N}\) (finite or not) satisfying the Robinson conditions. The transformations employed in the proof are described in a constructive way and are Abelian group extensions of adding machines. If, in addition, \(A\) is finite, then \(T\) can be chosen to be a Morse automorphism over a finite cyclic group.

Reviewer: V.A.Kaĭmanovich (Edinburgh) (MR 93f:28015)

MSC:

28D05

Measure-preserving transformations

Keywords:

multiplicity function; ergodic group extensions of rotations; spectral multiplicities; ergodic measure-preserving transformation; Abelian group extensions; adding machines; Morse automorphism

Citations:

Zbl 0678.28009