{
  "id": "1304.4556",
  "version": "v2",
  "published": "2013-04-16T19:04:25.000Z",
  "updated": "2013-05-16T14:52:11.000Z",
  "title": "Central limit theorem for commutative semigroups of toral endomorphisms",
  "authors": [
    "Guy Cohen",
    "Jean-Pierre Conze"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $\\Cal S$ be an abelian finitely generated semigroup of endomorphisms of a probability space $(\\Omega, {\\Cal A}, \\mu)$, with $(T_1, ..., T_d)$ a system of generators in ${\\Cal S}$. Given an increasing sequence of domains $(D_n) \\subset \\N^d$, a question is the convergence in distribution of the normalized sequence $|D_n|^{-\\frac12} \\sum_{{\\k} \\, \\in D_n} \\, f \\circ T^{\\,{\\k}}$, for $f \\in L^2_0(\\mu)$, where $T^{\\k}= T_1^{k_1} ... T_d^{k_d}$, ${\\k}= (k_1, ..., k_d) \\in {\\N}^d$. After a preliminary spectral study when the action of $\\Cal S$ has a Lebesgue spectrum, we consider $\\N^d$- or $\\Z^d$-actions given by commuting toral automorphisms or endomorphisms on $\\T^\\rho$, $\\rho \\geq 1$. For a totally ergodic action by automorphisms, we show a CLT for the above normalized sequence or other summation methods like barycenters, as well as a criterion of non-degeneracy of the variance, when $f$ is regular on the torus. A CLT is also proved for some semigroups of endomorphisms. Classical results on the existence and the construction of such actions by automorphisms are recalled.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-05-16T14:52:11.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "central limit theorem",
      "toral endomorphisms",
      "commutative semigroups",
      "normalized sequence",
      "preliminary spectral study"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1304.4556C"
    }
  }
}