{
  "id": "2003.12014",
  "version": "v1",
  "published": "2020-03-18T10:07:49.000Z",
  "updated": "2020-03-18T10:07:49.000Z",
  "title": "Coarse spaces, ultrafilters and dynamical systems",
  "authors": [
    "Igor Protasov"
  ],
  "comment": "Keywords: Coarse spaces, balleans, ultrafilters, dynamical systems",
  "categories": [
    "math.GN"
  ],
  "abstract": "For a coarse space $(X, \\mathcal{E})$, $X^\\sharp$ denotes the set of all unbounded ultrafilters on $X$ endowed with the parallelity relation: $p||q$ if there exists $E \\in \\mathcal{E} $ such that $ E[P]\\in q $ for each $P\\in p$. If $(X, \\mathcal{E})$ is finitary then there exists a group $G $ of permutations of $X$ such that the coarse structure $\\mathcal{E}$ has the base $\\{\\{ (x,gx): x\\in X$, $g\\in F\\}: F\\in [G]^{<\\omega}, \\ id \\in F \\}.$ We survey and analyze interplays between $(X, \\mathcal{E})$, $X^\\sharp$ and the dynamical system $(G, X^\\sharp)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-18T10:07:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "coarse space",
      "dynamical system",
      "parallelity relation",
      "coarse structure",
      "analyze interplays"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}