{
  "id": "math/0602083",
  "version": "v1",
  "published": "2006-02-05T14:58:41.000Z",
  "updated": "2006-02-05T14:58:41.000Z",
  "title": "Ergodic Transformations of the Space of $p$-adic Integers",
  "authors": [
    "Vladimir Anashin"
  ],
  "comment": "To be published in Proceedings of the 2-nd Int'l Conference on p-adic Mathematical Physics (15-25 Sept., 2005, Belgrade)",
  "journal": "AIP Conference Proceedings, vol. 826, pp.3--24, 2006",
  "categories": [
    "math.DS",
    "math.NT"
  ],
  "abstract": "Let $\\mathcal L_1$ be the set of all mappings $f\\colon\\Z_p\\Z_p$ of the space of all $p$-adic integers $\\Z_p$ into itself that satisfy Lipschitz condition with a constant 1. We prove that the mapping $f\\in\\mathcal L_1$ is ergodic with respect to the normalized Haar measure on $\\Z_p$ if and only if $f$ induces a single cycle permutation on each residue ring $\\Z/p^k\\Z$ modulo $p^k$, for all $k=1,2,3,...$. The multivariate case, as well as measure-preserving mappings, are considered also. Results of the paper in a combination with earlier results of the author give explicit description of ergodic mappings from $\\mathcal L_1$. This characterization is complete for $p=2$. As an application we obtain a characterization of polynomials (and certain locally analytic functions) that induce ergodic transformations of $p$-adic spheres. The latter result implies a solution of a problem (posed by A.~Khrennikov) about the ergodicity of a perturbed monomial mapping on a sphere.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-02-05T14:58:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A25",
      "37A05",
      "11S80",
      "05.20.-y",
      "02.30.Cj",
      "02.30.Sa",
      "02.10.De"
    ],
    "keywords": [
      "adic integers",
      "satisfy lipschitz condition",
      "induce ergodic transformations",
      "single cycle permutation",
      "normalized haar measure"
    ],
    "tags": [
      "conference paper",
      "journal article"
    ],
    "publication": {
      "publisher": "AIP",
      "journal": "AIP Conf. Proc."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006AIPC..826....3A"
    }
  }
}