{
  "id": "1610.01438",
  "version": "v1",
  "published": "2016-10-05T14:25:36.000Z",
  "updated": "2016-10-05T14:25:36.000Z",
  "title": "On conservative sequences and their application to ergodic multiplier problems",
  "authors": [
    "Madeleine Elyze",
    "Alexander Kastner",
    "Juan Ortiz Rhoton",
    "Vadim Semenov",
    "Cesar E. Silva"
  ],
  "comment": "26 pages, 3 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "The \\emph{conservative sequence} of a set $A$ under a transformation $T$ is the set of all $n \\in \\mathbb{Z}$ such that $T^n A \\cap A \\not = \\varnothing$. By studying these sequences, we prove that given any countable collection of nonsingular transformations with no finite invariant measure $\\{T_i\\}$, there exists a rank-one transformation $S$ such that $T_i \\times S$ is not ergodic for all $i$. Moreover, $S$ can be chosen to be rigid or have infinite ergodic index. We establish similar results for $\\mathbb{Z}^d$ actions and flows. Then, we find sufficient conditions on rank-one transformations $T$ that guarantee the existence of a rank-one transformation $S$ such that $T \\times S$ is ergodic, or, alternatively, conditions that guarantee that $T \\times S$ is conservative but not ergodic. In particular, the infinite Chac\\'on transformation satisfies both conditions. Finally, for a given ergodic transformation $T$, we study the Baire categories of the sets $E(T)$, $\\bar{E}C(T)$ and $\\bar{C}(T)$ of transformations $S$ such that $T \\times S$ is ergodic, ergodic but not conservative, and conservative, respectively.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-05T14:25:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "ergodic multiplier problems",
      "conservative sequences",
      "rank-one transformation",
      "infinite chacon transformation satisfies",
      "application"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}