{
  "id": "2007.07036",
  "version": "v1",
  "published": "2020-07-14T13:49:20.000Z",
  "updated": "2020-07-14T13:49:20.000Z",
  "title": "On periods of Herman rings and relevant poles",
  "authors": [
    "Subhasis Ghora",
    "Tarakanta Nayak"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "Possible periods of Herman rings are studied for general meromorphic functions with at least one omitted value. A pole is called $H$-relevant for a Herman ring $H$ of such a function $f$ if it is surrounded by some Herman ring of the cycle containing $H$. In this article, a lower bound on the period $p$ of a Herman ring $H$ is found in terms of the number of $H$-relevant poles, say $h$. More precisely, it is shown that $p\\geq \\frac{h(h+1)}{2}$ whenever $f^j(H)$, for some $j$, surrounds a pole as well as the set of all omitted values of $f$. It is proved that $p \\geq \\frac{h(h+3)}{2}$ in the other situation. Sufficient conditions are found under which equalities hold. It is also proved that if an omitted value is contained in the closure of an invariant or a two periodic Fatou component then the function does not have any Herman ring.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-14T13:49:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "herman ring",
      "relevant poles",
      "omitted value",
      "general meromorphic functions",
      "periodic fatou component"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}