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March 16, 2026Sbornik Mathematics2 citations

Spectra and joint dynamics of Poisson suspensions over rank-one automorphisms

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VRV. V. Ryzhikov

Key Points

  • The research investigates the spectral dynamics of unitary operators and their tensor powers, focusing on singular and absolutely continuous spectra.
  • Identified unitary operators with specific spectral properties across tensor powers.
  • Established conditions for sequences influencing spectral characteristics.
  • Analyzed mixing automorphisms in Poisson suspensions with zero entropy.
  • Demonstrated that certain tensor powers have a singular spectrum while subsequent powers may be absolutely continuous.
  • Found that diverging sequences influence spectral behavior, leading to complex interactions between automorphisms.
  • Showed existence of mixing automorphisms where intersections of sets remain empty across iterations.

Abstract

For each integer n>1 a unitary operator of dynamical origin is found such that its nth tensor power has a singular spectrum, but the spectrum of the (n+1) st power is absolutely continuous. For any sequences p (n) and q (n) such that p (n+1) - p (n) + and q (n+1) - q (n) + there exist a set C and automorphisms S and T with simple singular spectra such that the sequence ₍=₁^N (S^ p (n) C T^ q (n) C) /N is divergent. In the class of Poisson suspensions with zero entropy there exist mixing automorphisms S and T such that for some set D of positive measure, SⁿD TⁿD= for all n>0. Bibliography: 23 titles.

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Cite This Study

V. V. Ryzhikov (2026) studied this question.

synapsesocial.com/papers/69b79e7c8166e15b153abea6https://doi.org/10.4213/sm10228e
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Also Consider

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