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Spectral multiplicity for powers of weakly mixing automorphisms. (English. Russian original) Zbl 1259.37005
Sb. Math. 203, No. 7, 1065-1076 (2012); translation from Mat. Sb. 203, No. 7, 149-160 (2012).
Author’s abstract: We study the behaviour of the maximal spectral multiplicity $$\mathfrak{m}(R^n)$$ for the powers of a weakly mixing automorphism $$R$$. For some particular infinite sets $$A$$, we show that there exists a weakly mixing rank-one automorphism $$R$$ such that $$\mathfrak{m}(R^n)=n$$ and $$\mathfrak{m}(R^{n+1})=1$$ for all positive integers $$n\in A$$. Moreover, the cardinality $$\text{cardm}(R^n)$$ of the set of spectral multiplicities for the power $$R^n$$ is shown to satisfy the conditions $$\text{cardm}(R^{n+1})=1$$ and $$\text{cardm}(R^n)=2^{m(n)}, m(n)\to\infty, n\in A$$. We also construct another weakly mixing automorphism $$R$$ with the following properties: all powers $$R^{n}$$ have homogeneous spectra and the set of limit points of the sequence $$\{\mathfrak{m}(R^n)/n:n\in \mathbb N \}$$ is infinite.

##### MSC:
 37A30 Ergodic theorems, spectral theory, Markov operators 28D05 Measure-preserving transformations 47A35 Ergodic theory of linear operators
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