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

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.

### MSC:

 28D05 Measure-preserving transformations

Zbl 0678.28009
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