{
  "id": "2307.06653",
  "version": "v1",
  "published": "2023-07-13T09:50:04.000Z",
  "updated": "2023-07-13T09:50:04.000Z",
  "title": "Veech's Theorem of $G$ acting freely on $G^{\\textrm{LUC}}$ and Structure Theorem of a.a. flows",
  "authors": [
    "Xiongping Dai",
    "Hailan Liang",
    "Zhengyu Yin"
  ],
  "comment": "54 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "Veech's Theorem claims that if $G$ is a locally compact\\,(LC) Hausdorff topological group, then it may act freely on $G^{\\textrm{LUC}}$. We prove Veech's Theorem for $G$ being only locally quasi-totally bounded, not necessarily LC. And we show that the universal a.a. flow is the maximal almost 1-1 extension of the universal minimal a.p. flow and is unique up to almost 1-1 extensions. In particular, every endomorphism of Veech's hull flow induced by an a.a. function is almost 1-1; for $G=\\mathbb{Z}$ or $\\mathbb{R}$, $G$ acts freely on its canonical universal a.a. space. Finally, we characterize Bochner a.a. functions on a LC group $G$ in terms of Bohr a.a. function on $G$ (due to Veech 1965 for the special case that $G$ is abelian, LC, $\\sigma$-compact, and first countable).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-13T09:50:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B05"
    ],
    "keywords": [
      "structure theorem",
      "veechs hull flow",
      "veechs theorem claims",
      "hausdorff topological group",
      "lc group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 54,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}