{
  "id": "1511.00172",
  "version": "v1",
  "published": "2015-10-31T20:51:13.000Z",
  "updated": "2015-10-31T20:51:13.000Z",
  "title": "Broadband nature of power spectra for intermittent Maps with summable and nonsummable decay of correlations",
  "authors": [
    "Georg A. Gottwald",
    "Ian Melbourne"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "We present results on the broadband nature of the power spectrum $S(\\omega)$, $\\omega\\in(0,2\\pi)$, for a large class of nonuniformly expanding maps with summable and nonsummable decay of correlations. In particular, we consider a class of intermittent maps $f:[0,1]\\to[0,1]$ with $f(x)\\approx x^{1+\\gamma}$ for $x\\approx 0$, where $\\gamma\\in(0,1)$. Such maps have summable decay of correlations when $\\gamma\\in(0,\\frac12)$, and $S(\\omega)$ extends to a continuous function on $[0,2\\pi]$ by the classical Wiener-Khintchine Theorem. We show that $S(\\omega)$ is typically bounded away from zero for H\\\"older observables. Moreover, in the nonsummable case $\\gamma\\in[\\frac12,1)$, we show that $S(\\omega)$ is defined almost everywhere with a continuous extension $\\tilde S(\\omega)$ defined on $(0,2\\pi)$, and $\\tilde S(\\omega)$ is typically nonvanishing.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-31T20:51:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "intermittent maps",
      "broadband nature",
      "power spectrum",
      "nonsummable decay",
      "correlations"
    ],
    "publication": {
      "doi": "10.1088/1751-8113/49/17/174003",
      "journal": "Journal of Physics A Mathematical General",
      "year": 2016,
      "month": "Apr",
      "volume": 49,
      "number": 17,
      "pages": 174003
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016JPhA...49q4003G"
    }
  }
}