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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.

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