{
  "id": "math/0601622",
  "version": "v1",
  "published": "2006-01-25T19:31:02.000Z",
  "updated": "2006-01-25T19:31:02.000Z",
  "title": "Structure Theorem for (d,g,h)-Maps",
  "authors": [
    "Alex V. Kontorovich",
    "Yakov G. Sinai"
  ],
  "comment": "9 pages",
  "journal": "Bulletin of the Brazilian Mathematical Society, Volume 33, Issue 2, Jul 2002, Pages 213 - 224",
  "categories": [
    "math.NT",
    "math.PR"
  ],
  "abstract": "The (3x+1)-Map, T, acts on the set, Pi, of positive integers not divisible by 2 or 3. It is defined by T(x) = (3x+1)/2^k, where k is the largest integer for which T(x) is an integer. The (3x+1)-Conjecture asks if for every x in Pi there exists an integer, n, such that T^n (x) = 1. The Statistical (3x+1)-Conjecture asks the same question, except for a subset of Pi of density 1. The Structure Theorem proven in \\cite{sinai} shows that infinity is in a sense a repelling point, giving some reasons to expect that the (3x+1)-Conjecture may be true. In this paper, we present the analogous theorem for some generalizations of the (3x+1)-Map, and expand on the consequences derived in \\cite{sinai}. The generalizations we consider are determined by positive coprime integers, d and g, with g > d >= 2, and a periodic function, h(x). The map T is defined by the formula T(x) = (gx+h(gx))/d^k, where k is again the largest integer for which T(x) is an integer. We prove an analogous Structure Theorem for (d,g,h)-Maps, and that the probability distribution corresponding to the density converges to the Wiener measure with the drift log(g) - d/(d-1)log(d) and positive diffusion constant. This shows that it is natural to expect that typical trajectores return to the origin if log(g) - d/(d-1) log(d) <0 and escape to infinity otherwise.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-01-25T19:31:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B83"
    ],
    "keywords": [
      "largest integer",
      "structure theorem proven",
      "typical trajectores return",
      "coprime integers",
      "drift log"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......1622K"
    }
  }
}