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

TL;DR

This paper constructs specific automorphisms with mixed spectral properties and explores their joint dynamics, providing answers to questions about spectral types and intersection behaviors in ergodic theory.

Contribution

It introduces automorphisms with tensor powers exhibiting singular and absolutely continuous spectra, and addresses intersection properties of automorphisms with divergent sums, answering open questions.

Findings

Existence of operators with tensor powers switching between singular and absolutely continuous spectra.

Construction of mixing zero entropy automorphisms with disjoint orbit intersections.

Demonstration of divergent sums of measure intersections for certain automorphisms.

Abstract

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