## Inverse limit of $$M$$-cocyles and applications.(English)Zbl 0918.28016

The author provides the final step in the solution to Mentzen’s problem: Given a pair of positive integers $$(m,r)$$, $$m\leq r\leq\infty$$, does there exist an ergodic automorphism $$T$$ on a Lebesgue probability space $$(X,{\mathcal B},\mu)$$, having rank $$r$$ and maximal spectral multiplicity $$m$$? A number of authors have contributed to the solution of this problem. In particular, J. Kwiatkowski and Y. Lacroix [J. Anal. Math. 71, 205-235 (1997; Zbl 0894.28008)] showed that any pair $$(m,r)$$, $$m\leq r<\infty$$ is obtainable. In this paper, the author shows how to realize the pair $$(m,\infty)$$ for any $$m\geq 1$$. In the second part of the paper, for each $$r\geq 1$$ and $$0< b<1$$ there is constructed an ergodic automorphism having simple spectrum, rank $$r$$, infinite essential centralizer and covering number equal to $$b$$.

### MSC:

 28D05 Measure-preserving transformations 54H20 Topological dynamics (MSC2010)

Zbl 0894.28008
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