{
  "id": "2307.02134",
  "version": "v1",
  "published": "2023-07-05T09:24:16.000Z",
  "updated": "2023-07-05T09:24:16.000Z",
  "title": "Invariant measures for $\\mathscr{B}$-free systems revisited",
  "authors": [
    "Aurelia Dymek",
    "Joanna Kułaga-Przymus",
    "Daniel Sell"
  ],
  "comment": "28 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "For $ \\mathscr{B} \\subseteq \\mathbb{N} $, the $ \\mathscr{B} $-free subshift $ X_{\\eta} $ is the orbit closure of the characteristic function of the set of $ \\mathscr{B} $-free integers. We show that many results about invariant measures and entropy, previously only known for the hereditary closure of $ X_{\\eta} $, have their analogues for $ X_{\\eta} $ as well. In particular, we settle in the affirmative a conjecture of Keller about a description of such measures ([Keller, G. Generalized heredity in $\\mathcal B$-free systems. Stoch. Dyn. 21, 3 (2021), Paper No. 2140008]). A central assumption in our work is that $ \\eta^{*} $ (the Toeplitz sequence that generates the unique minimal component of $ X_{\\eta} $) is regular. From this we obtain natural periodic approximations that we frequently use in our proofs to bound the elements in $ X_{\\eta} $ from above and below.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-05T09:24:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28C10",
      "37A05",
      "37A44",
      "37B40",
      "37B10"
    ],
    "keywords": [
      "free systems",
      "invariant measures",
      "unique minimal component",
      "natural periodic approximations",
      "characteristic function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}