{
  "id": "1609.02529",
  "version": "v1",
  "published": "2016-09-08T18:49:04.000Z",
  "updated": "2016-09-08T18:49:04.000Z",
  "title": "Pointwise convergence of some multiple ergodic averages",
  "authors": [
    "Sebastián Donoso",
    "Wenbo Sun"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "We show that for every ergodic system $(X,\\mu,T_1,\\ldots,T_d)$ with commuting transformations, the average \\[\\frac{1}{N^{d+1}} \\sum_{0\\leq n_1,\\ldots,n_d \\leq N-1} \\sum_{0\\leq n\\leq N-1} f_1(T_1^n \\prod_{j=1}^d T_j^{n_j}x)f_2(T_2^n \\prod_{j=1}^d T_j^{n_j}x)\\cdots f_d(T_d^n \\prod_{j=1}^d T_j^{n_j}x). \\] converges for $\\mu$-a.e. $x\\in X$ as $N\\to\\infty$. If $X$ is distal, we prove that the average \\[\\frac{1}{N}\\sum_{i=0}^{N} f_1(T_1^nx)f_2(T_2^nx)\\cdots f_d(T_d^nx) \\] converges for $\\mu$-a.e. $x\\in X$ as $N\\to\\infty$. We also establish the pointwise convergence of averages along cubical configurations arising from a system commuting transformations. Our methods combine the existence of sated and magic extensions introduced by Austin and Host respectively with ideas on topological models by Huang, Shao and Ye.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-08T18:49:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A30",
      "54H20"
    ],
    "keywords": [
      "multiple ergodic averages",
      "pointwise convergence",
      "magic extensions",
      "ergodic system",
      "system commuting transformations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}