{
  "id": "1701.04345",
  "version": "v1",
  "published": "2017-01-16T16:13:57.000Z",
  "updated": "2017-01-16T16:13:57.000Z",
  "title": "Over Recurrence for Mixing Transformations",
  "authors": [
    "Terrence Adams"
  ],
  "comment": "18 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "We show that every invertible strong mixing transformation on a Lebesgue space has strictly over-recurrent sets. Also, we give an explicit procedure for constructing strong mixing transformations with no under-recurrent sets. This answers both parts of a question of V. Bergelson. We define $\\epsilon$-over-recurrence and show that given $\\epsilon > 0$, any ergodic measure preserving invertible transformation (including discrete spectrum) has $\\epsilon$-over-recurrent sets of arbitrarily small measure. Discrete spectrum transformations and rotations do not have over-recurrent sets, but we construct a weak mixing rigid transformation with strictly over-recurrent sets.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-16T16:13:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A25",
      "28D05"
    ],
    "keywords": [
      "strictly over-recurrent sets",
      "recurrence",
      "discrete spectrum transformations",
      "weak mixing rigid transformation",
      "ergodic measure preserving invertible transformation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}