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

2024/11/04 by Valery V. Ryzhikov, Ryzhikov, Valery V. · 2 citations
Engineering · Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Control and Dynamics of Mobile Robots #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2411.02024

openalex publication_date 2024/11/04 · openalex created_date 2024/11/15 · openalex updated_date 2026/07/28

Abstract

For every natural n>1, there is an operator T of dynamical origin such that its tensor power T^{⊗ n} has singular spectrum, and T^{⊗ (n+1)} has absolutely continuous one. For a set D of positive measure there are mixing zero entropy automorphisms S,T such that SⁿD∩ TⁿD=\varnothing for all n>0. The following answers, in particular, Frantzikinakis-Host's question. If p(n+1)- p(n), q(n+1)- q(n) → +∞, then there is a divergent sequence ∑_n=1^N μ(S ^p(n)C∩ T ^q(n)C)/N for some set C and automorphisms S,T of simple singular spectrum.

Cited by

• Continuum pairwise disjoint automorphisms with Lebesgue spectrum
• Counterexamples to double recurrence for non-commuting deterministic transformations

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