{
  "id": "2312.12562",
  "version": "v1",
  "published": "2023-12-19T19:57:07.000Z",
  "updated": "2023-12-19T19:57:07.000Z",
  "title": "Almost automorphic systems have invariant measures",
  "authors": [
    "María Isabel Cortez",
    "Jaime Gómez"
  ],
  "comment": "23 pages",
  "categories": [
    "math.DS",
    "math.GR"
  ],
  "abstract": "Let $G$ be a non-amenable countable group. We show that every almost automorphic $G$-action on a compact Hausdorff space, with a maximal equicontinuous factor whose phase space is a Cantor set, admits invariant probability measures (this partially answers a question posed by Veech). In particular, every Toeplitz $G$-subshift has a non-empty space of invariant measures, meaning that this family of subshifts is not a test for amenability for countable groups. We prove that almost one-to-one extensions without measures ensure the existence of symbolic almost one-to-one extensions with equal characteristics. As a consequence, we obtain the most general result of this paper. Finally, as a corollary of our results, we deduce that the class of Toeplitz subshifts is not dense in the space of infinite transitive subshifts of $\\Sigma^G$, unlike $G=\\mathbb{Z}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-19T19:57:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B05",
      "37B10"
    ],
    "keywords": [
      "invariant measures",
      "automorphic systems",
      "one-to-one extensions",
      "admits invariant probability measures",
      "compact hausdorff space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}