{
  "id": "1511.06868",
  "version": "v1",
  "published": "2015-11-21T11:35:33.000Z",
  "updated": "2015-11-21T11:35:33.000Z",
  "title": "Decay of correlation for expanding toral endomorphisms",
  "authors": [
    "Ai Hua Fan"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $A$ be an expanding endomorphism on the torus ${\\Bbb T}^d = {\\Bbb R}^d /{\\Bbb Z}^d$ with its smallest eigenvalue $\\lambda >1$. Consider the ergodic system $({\\Bbb T}^d, A, \\mu)$ where $\\mu$ is Haar measure. We prove that the correlation $\\rho_{f, g}(n)$ of a pair of functions $f, g \\in L^2(\\mu)$ is controlled by the modulus of $L^2$-continuity $\\Omega_{f, 2}(\\lambda^{-n})$ and that the estimate is to some extent optimal. We also prove the central limit theorem for the stationary process $f(A^n x)$ defined by a function $f$ satisfying $\\Sigma_n \\Omega_{f,2}(\\lambda^{-n}) <\\infty$. An application is given to the Ulam-von Neumann system.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-21T11:35:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "expanding toral endomorphisms",
      "correlation",
      "ulam-von neumann system",
      "central limit theorem",
      "smallest eigenvalue"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}