{
  "id": "2202.09336",
  "version": "v1",
  "published": "2022-02-18T18:04:59.000Z",
  "updated": "2022-02-18T18:04:59.000Z",
  "title": "Absolute continuity and singularity of spectra for flows $T_t\\otimes T_{at}$",
  "authors": [
    "Valery V. Ryzhikov"
  ],
  "comment": "in Russian language",
  "categories": [
    "math.DS"
  ],
  "abstract": "Answering the question of V.I. Oseledets, we present a random variable $\\xi$ such that the sum $\\xi(x)+a\\xi(y)$ 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 $\\xi$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-18T18:04:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "absolute continuity",
      "singularity",
      "simple singular spectra",
      "infinite lebesgue space",
      "singular distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "ru",
      "license": "arXiv",
      "status": "editable"
    }
  }
}