{
  "id": "1303.0374",
  "version": "v1",
  "published": "2013-03-02T11:08:44.000Z",
  "updated": "2013-03-02T11:08:44.000Z",
  "title": "Minimal sets of fibre-preserving maps in graph bundles",
  "authors": [
    "Sergii Kolyada",
    "Ľubomír Snoha",
    "Sergei Trofimchuk"
  ],
  "comment": "43 pages, 2 figures, 34 references, submitted",
  "categories": [
    "math.DS"
  ],
  "abstract": "Topological structure of minimal sets is studied for a dynamical system $(E,F)$ given by a fibre-preserving, in general non-invertible, continuous selfmap $F$ of a graph bundle $E$. These systems include, as a very particular case, quasiperiodically forced circle homeomorphisms. Let $M$ be a minimal set of $F$ with full projection onto the base space $B$ of the bundle. We show that $M$ is nowhere dense or has nonempty interior depending on whether the set of so called endpoints of $M$ is dense in $M$ or is empty. If $M$ is nowhere dense, we prove that either a typical fibre of $M$ is a Cantor set, or there is a positive integer $N$ such that a typical fibre of $M$ has cardinality $N$. If $M$ has nonempty interior we prove that there is a positive integer $m$ such that a typical fibre of $M$, in fact even each fibre of $M$ over a \\emph{dense open} set $\\mathcal O \\subseteq B$, is a disjoint union of $m$ circles. Moreover, we show that each of the fibres of $M$ over $B\\setminus \\mathcal O$ is a union of circles properly containing a disjoint union of $m$ circles. Surprisingly, some of the circles in such \"non-typical\" fibres of $M$ may intersect. We also give sufficient conditions for $M$ to be a sub-bundle of $E$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-03-02T11:08:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54H20",
      "37B05"
    ],
    "keywords": [
      "minimal set",
      "graph bundle",
      "fibre-preserving maps",
      "typical fibre",
      "disjoint union"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1303.0374K"
    }
  }
}