Abstract
Given disjoint countable dense subsets \(C\) and \(D\) of the half-line \((1,+\infty)\), there exists a flow \(T_t\) preserving a sigma-finite measure and such that all automorphisms \(T_1\otimes T_{c}\) with \(c\in C\) have simple singular spectrum and all automorphisms \(T_1\otimes T_{d}\) with \(d\in D\) have Lebesgue spectrum of countable multiplicity.
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References
Z. I. Bezhaeva and V. I. Oseledets, Zap. Nauchn. Sem. POMI, 326 (2005), 28–47; English transl.: J. Math. Sci. (N. Y.), 140:3 (2007), 357–368.
Z. I. Bezhaeva and V. I. Oseledets, Funkts. Anal. Ego Prilozh., 44:2 (2010), 3–13; English transl. Funct. Anal. Appl., 44:2 (2010), 83–91.
V. V. Ryzhikov, Mat. Zametki, 106:6 (2019), 894–903; Math. Notes, 106:6 (2019), 957–965.
Acknowledgments
The author thanks B. Weiss for discussions of V. I. Oseledets’ problem.
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Translated from Funktsional'nyi Analiz i ego Prilozheniya, 2022, Vol. 56, pp. 88–92 https://doi.org/10.4213/faa3986.
Translated by O. V. Sipacheva
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Ryzhikov, V.V. Absolute Continuity and Singularity of Spectra for the Flows \(T_t\otimes T_{at}\). Funct Anal Its Appl 56, 225–228 (2022). https://doi.org/10.1134/S0016266322030066
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DOI: https://doi.org/10.1134/S0016266322030066