{
  "id": "2301.12384",
  "version": "v1",
  "published": "2023-01-29T07:41:16.000Z",
  "updated": "2023-01-29T07:41:16.000Z",
  "title": "Persistent Shadowing For Actions Of Some Finitely Generated Groups and Related Measures",
  "authors": [
    "Ali Barzanouni"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "In this paper, $\\varphi:G\\times X\\to X$ is a continuous action of finitely generated group $G$ on compact metric space $(X, d)$ without isolated point. We introduce the notion of persistent shadowing property for $\\varphi:G\\times X\\to X$ and study it via measure theory. Indeed, we introduce the notion of compatibility the Borel probability measure $\\mu$ with respect persistent shadowing property of $\\varphi:G\\times X\\to X$ and denote it by $\\mu\\in\\mathcal{M}_{PSh}(X, \\varphi)$. We show $\\mu\\in\\mathcal{M}_{PSh}(X, \\varphi)$ if and only if $supp(\\mu)\\subseteq PSh(\\varphi)$, where $PSh(\\varphi)$ is the set of all persistent shadowable points of $\\varphi$. This implies that if every non-atomic Borel probability measure $\\mu$ is compatible with persistent shadowing property for $\\varphi:G\\times X\\to X$, then $\\varphi$ does have persistent shadowing property. We prove that $\\overline{PSh(\\varphi)}=PSh(\\varphi)$ if and only if $\\overline{\\mathcal{M}_{PSh}(X, \\varphi)}= \\mathcal{M}_{PSh}(X, \\varphi)$. Also, $\\mu(\\overline{PSh(\\varphi)})=1$ if and only if $\\mu\\in\\overline{\\mathcal{M}_{PSh}(X, \\varphi)}$. Finally, we show that $\\overline{\\mathcal{M}_{PSh}(X, \\varphi)}=\\mathcal{M}(X)$ if and only if $\\overline{PSh(\\varphi)}=X$. For study of persistent shadowing property, we introduce the notions of uniformly $\\alpha$-persistent point, uniformly $\\beta$-persistent point and recall notions of shadowing property, $\\alpha$-persistent, $\\beta$-persistent and we give some further results about them.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-29T07:41:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finitely generated group",
      "related measures",
      "non-atomic borel probability measure",
      "persistent point",
      "respect persistent shadowing property"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}