{
  "id": "2208.11675",
  "version": "v1",
  "published": "2022-08-24T17:12:46.000Z",
  "updated": "2022-08-24T17:12:46.000Z",
  "title": "Collatz map as a power bounded nonsingular transformation",
  "authors": [
    "Idris Assani"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $T$ be the map defined on $\\N$ by $T(n) = \\frac{n}{2} $ if $n$ is even and by $T(n) = \\frac{3n+1}{2}$ if $n$ is odd. Consider the dynamical system $(\\N, 2^{\\N}, \\nu, T)$ where $\\nu$ is a finite measure equivalent to the counting measure. Define the operator $U : L^1(\\nu) \\rightarrow L^1(\\nu) $ by $Uf = f\\circ T. $ We prove that for each $n\\in \\N$ the set $\\{T^k(n) : k\\in\\N\\}$ is bounded. We show that the Collatz conjecture is equivalent to the existence of a finite measure $\\nu$ on $(\\N, 2^{\\N})$ making the operator $Vf = f\\circ T$ power bounded in $L^1(\\nu).$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-24T17:12:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B75",
      "37A40"
    ],
    "keywords": [
      "power bounded nonsingular transformation",
      "collatz map",
      "finite measure equivalent",
      "collatz conjecture",
      "counting measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}