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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)

Citations:

Zbl 0894.28008
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