Spectra and joint dynamics of Poisson suspensions over rank-one automorphisms
Abstract
For each integer $n>1$ a unitary operator of dynamical origin is found such that its $n$th 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) \to+\infty$ and $q(n+1)- q(n)\to+\infty$ there exist a set $C$ and automorphisms $S$ and $T$ with simple singular spectra such that the sequence $\sum_{n=1}^{N} \mu(S^{ p(n)}C\cap T^{ q(n)}C)/N$ 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, $S^nD\cap T^nD=\varnothing$ for all $n>0$. Bibliography: 23 titles.
Fields of Study & Subjects
Advanced Differential Equations and Dynamical Systems
Mathematical Dynamics and Fractals
Holomorphic and Operator Theory
Publication details
- Journal / source
- Sbornik Mathematics
- Publisher
- Steklov Mathematical Institute
- Volume
- 217
- Issue
- 1
- Pages
- 91–104
- Publication date
- 1 January 2026
- Language
- EN