{
  "id": "2407.07214",
  "version": "v1",
  "published": "2024-07-09T20:17:56.000Z",
  "updated": "2024-07-09T20:17:56.000Z",
  "title": "Trichotomy for the orbits of a hypercyclic operator on a Banach space",
  "authors": [
    "Jian Li"
  ],
  "comment": "12 papers,to appear in Proc. Amer. Math. Soc",
  "doi": "10.1090/proc/16974",
  "categories": [
    "math.FA",
    "math.DS"
  ],
  "abstract": "We obtain a trichotomy for the orbits of a hypercyclic operator $T$ on a separable Banach space $X$: (1) every vector is mean asymptotic to zero; (2) generic vectors are absolutely mean irregular; (3) every hypercyclic vector is mean divergent to infinity. Examples of weighted backward shifts on $\\ell^p$ show that all three cases can happen.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-09T20:17:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A16",
      "37B05"
    ],
    "keywords": [
      "hypercyclic operator",
      "trichotomy",
      "mean asymptotic",
      "separable banach space",
      "generic vectors"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "AMS"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}