{
  "id": "1910.00369",
  "version": "v1",
  "published": "2019-10-01T13:24:56.000Z",
  "updated": "2019-10-01T13:24:56.000Z",
  "title": "On the fractional susceptibility function of piecewise expanding maps",
  "authors": [
    "M. Aspenberg",
    "V. Baladi",
    "J. Leppänen",
    "T. Persson"
  ],
  "categories": [
    "math.DS",
    "math-ph",
    "math.MP",
    "nlin.CD"
  ],
  "abstract": "We associate to a perturbation $(f_t)$ of a (stably mixing) piecewise expanding unimodal map $f_0$ a two-variable fractional susceptibility function $\\Psi_\\phi(\\eta, z)$, depending also on a bounded observable $\\phi$. For fixed $\\eta \\in (0,1)$, we show that the function $\\Psi_\\phi(\\eta, z)$ is holomorphic in a disc $D_\\eta\\subset \\mathbb{C}$ centered at zero of radius $>1$, and that $\\Psi_\\phi(\\eta, 1)$ is the Marchaud fractional derivative of order $\\eta$ of the function $t\\mapsto \\mathcal{R}_\\phi(t):=\\int \\phi(x)\\, d\\mu_t$, at $t=0$, where $\\mu_t$ is the unique absolutely continuous invariant probability measure of $f_t$. In addition, we show that $\\Psi_\\phi(\\eta, z)$ admits a holomorphic extension to the domain $\\{ (\\eta, z) \\in {\\mathbb{C}}^2\\mid 0<\\Re \\eta <1, \\, z \\in D_\\eta \\}$. Finally, if the perturbation $(f_t)$ is horizontal, we prove that $\\lim_{\\eta \\to 1}\\Psi_\\phi(\\eta, 1)=\\partial_t \\mathcal{R}_\\phi(t)|_{t=0}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-01T13:24:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37C30",
      "37C40",
      "37E05",
      "37A10"
    ],
    "keywords": [
      "piecewise expanding maps",
      "unique absolutely continuous invariant probability",
      "absolutely continuous invariant probability measure",
      "two-variable fractional susceptibility function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}