{
  "id": "1309.4315",
  "version": "v4",
  "published": "2013-09-17T14:06:10.000Z",
  "updated": "2014-06-19T20:03:33.000Z",
  "title": "Non-conventional ergodic averages for several commuting actions of an amenable group",
  "authors": [
    "Tim Austin"
  ],
  "comment": "40 pages [Oct 28th 2013:] Former appendix moved to separate note: http://arxiv.org/abs/1310.6781 [Jun 19th 2014:] Updated following referee's suggestions",
  "categories": [
    "math.DS",
    "math.CO",
    "math.GR"
  ],
  "abstract": "Let $(X,\\mu)$ be a probability space, $G$ a countable amenable group and $(F_n)_n$ a left F\\o lner sequence in $G$. This paper analyzes the non-conventional ergodic averages \\[\\frac{1}{|F_n|}\\sum_{g \\in F_n}\\prod_{i=1}^d (f_i\\circ T_1^g\\cdots T_i^g)\\] associated to a commuting tuple of $\\mu$-preserving actions $T_1$, ..., $T_d:G\\curvearrowright X$ and $f_1$, ..., $f_d \\in L^\\infty(\\mu)$. We prove that these averages always converge in $\\|\\cdot\\|_2$, and that they witness a multiple recurrence phenomenon when $f_1 = \\ldots = f_d = 1_A$ for a non-negligible set $A\\subseteq X$. This proves a conjecture of Bergelson, McCutcheon and Zhang. The proof relies on an adaptation from earlier works of the machinery of sated extensions.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2014-06-19T20:03:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A25",
      "28D15"
    ],
    "keywords": [
      "non-conventional ergodic averages",
      "amenable group",
      "commuting actions",
      "multiple recurrence phenomenon",
      "probability space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1309.4315A"
    }
  }
}