{
  "id": "2303.11487",
  "version": "v1",
  "published": "2023-03-20T22:52:13.000Z",
  "updated": "2023-03-20T22:52:13.000Z",
  "title": "On the properties of the mean orbital pseudo-metric",
  "authors": [
    "Fangzhou Cai",
    "Dominik Kwietniak",
    "Jian Li",
    "Habibeh Pourmand"
  ],
  "journal": "Journal of Differential Equations 318 (2022), 1-19",
  "doi": "10.1016/j.jde.2022.02.019",
  "categories": [
    "math.DS"
  ],
  "abstract": "Given a topological dynamical system $(X,T)$, we study properties of the mean orbital pseudo-metric $\\bar E$ defined by \\[ \\bar E(x,y)= \\limsup_{n\\to\\infty } \\min_{\\sigma\\in S_n}\\frac{1}{n}\\sum_{k=0}^{n-1}d(T^k(x),T^{\\sigma(k)}(y)), \\] where $x,y\\in X$ and $S_n$ is the permutation group of $\\{0,1,\\ldots,n-1\\}$. Let $\\hat\\omega_T(x)$ denote the set of measures quasi-generated by a point $x\\in X$. We show that the map $x\\mapsto\\hat\\omega_T(x)$ is uniformly continuous if $X$ is endowed with the pseudo-metric $\\bar E$ and the space of compact subsets of the set of invariant measures is considered with the Hausdorff distance. We also obtain a new characterisation of $\\bar E$-continuity, which connects it to other properties studied in the literature, like continuous pointwise ergodicity introduced by Downarowicz and Weiss. Finally, we apply our results to reprove some known results on $\\bar E$-continuous and mean equicontinuous systems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-03-20T22:52:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "mean orbital pseudo-metric",
      "study properties",
      "mean equicontinuous systems",
      "compact subsets",
      "invariant measures"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}