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

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 ∑ₙ₌₁N μ(Sᵖ⁽ⁿ⁾C∩ Tq(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.