流$T_t\otimes T_{at}$谱的绝对连续性和奇异性
Absolute continuity and singularity of spectra for flows $T_t\otimes T_{at}$
摘要：
关于V.I. Oseledets，我们提出了一个随机变量$ξ$，使总和$ξ(x)+aξ(y)$有一个奇异分布的一组参数$a$密集在$(1, +\infty)$，但另一个密集的参数集，这个总和有一个绝对连续的分布。我们证明了以下断言：给定$C,D$，射线$(1,+\infty)$的可数非相交稠密子集，存在保测度流$T_t$（作用于无限Lebesgue空间），使得自同构$T_1\otimes T_{c}$对每个$c\in C$都有简单奇异谱，而$T_1\otimes T_{d}$对所有$d\in D$都有Lebesgue谱.该流的谱测度起着随机变量$ξ$的分布的作用。
英文摘要：
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 $(1, +\infty)$, 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 $(1,+\infty)$, there is a measure-preserving flow $T_t$ (acting on the infinite Lebesgue space) such that automorphisms $T_1\otimes T_{c}$ have simple singular spectra for every $c\in C$, and $T_1\otimes T_{d}$ have Lebesgue spectra for all $d\in D$. The spectral measure of this flow plays the role of the distribution of our random variable $ξ$.
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