{
  "id": "1411.6487",
  "version": "v1",
  "published": "2014-11-24T15:34:10.000Z",
  "updated": "2014-11-24T15:34:10.000Z",
  "title": "Almost everywhere convergence of ergodic series",
  "authors": [
    "Aihua Fan"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "We consider ergodic series of the form $\\sum_{n=0}^\\infty a_n f(T^n x)$ where $f$ is an integrable function with zero mean value with respect to a $T$-invariant measure $\\mu$. Under certain conditions on the dynamical system $T$, the invariant measure $\\mu$ and the function $f$, we prove that the series converges $\\mu$-almost everywhere if and only if $\\sum_{n=0}^\\infty |a_n|^2<\\infty$, and that in this case the sum of the convergent series is exponentially integrable and satisfies a Khintchine type inequality. We also prove that the system $\\{f\\circ T^n\\}$ is a Riesz system if and only if the spectral measure of $f$ is absolutely continuous with respect to the Lebesgue measure and the Radon-Nikodym derivative is bounded from above as well as from below by a constant. We check the conditions for Gibbs measures $\\mu$ relative to hyperbolic dynamics $T$ and for H\\\"{o}lder functions $f$. An application is given to the study of differentiability of the Weierstrass type functions $\\sum_{n=0}^\\infty a_n f(3^n x)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-11-24T15:34:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A30"
    ],
    "keywords": [
      "ergodic series",
      "convergence",
      "invariant measure",
      "weierstrass type functions",
      "khintchine type inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1411.6487F"
    }
  }
}