{
  "id": "1806.07413",
  "version": "v1",
  "published": "2018-06-19T18:20:28.000Z",
  "updated": "2018-06-19T18:20:28.000Z",
  "title": "Chaos in convolution operators on the space of entire functions of infinitely many complex variables",
  "authors": [
    "Blas M. Caraballo",
    "Vinícius V. Fávaro"
  ],
  "comment": "10 pages",
  "categories": [
    "math.FA",
    "math.DS"
  ],
  "abstract": "A classical result of Godefroy and Shapiro states that every nontrivial convolution operator on the space $\\mathcal{H}(\\mathbb{C}^n)$ of entire functions of several complex variables is hypercyclic. In sharp contrast with this result F\\'avaro and Mujica show that no translation operator on the space $\\mathcal{H}(\\mathbb{C}^\\mathbb{N})$ of entire functions of infinitely many complex variables is hypercyclic. In this work we study the linear dynamics of convolution operators on $\\mathcal{H}(\\mathbb{C}^\\mathbb{N})$. First we show that no convolution operator on $\\mathcal{H}(\\mathbb{C}^\\mathbb{N})$ is neither cyclic nor $n$-supercyclic for any positive integer $n$. After we study the notion of Li--Yorke chaos in non-metrizable topological vector spaces and we show that every nontrivial convolution operator on $\\mathcal{H}(\\mathbb{C}^\\mathbb{N})$ is Li--Yorke chaotic.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-19T18:20:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "complex variables",
      "entire functions",
      "nontrivial convolution operator",
      "li-yorke chaotic",
      "shapiro states"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}