Absolute continuity and singularity of spectra for flows T_t⊗ T_at

2022/02/18 by Valery V. Ryzhikov, Ryzhikov, Valery V.
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2202.09336

openalex publication_date 2022/02/18 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28

Abstract

Answering the question of V.I. Oseledets, we present a random variable ξ such that the sum ξ(x)+aξ(y) has a singular distribution for a set of parameters a dense in (1, +∞), but for another dense set of parameters, this sum has an absolutely continuous distribution. We prove the following assertion: given C,D, countable non-intersecting dense subsets of the ray (1,+∞), there is a measure-preserving flow T_t (acting on the infinite Lebesgue space) such that automorphisms T₁⊗ T_c have simple singular spectra for every c∈ C, and T₁⊗ T_d have Lebesgue spectra for all d∈ D. The spectral measure of this flow plays the role of the distribution of our random variable ξ.

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