{
  "id": "1709.02451",
  "version": "v1",
  "published": "2017-09-07T21:04:17.000Z",
  "updated": "2017-09-07T21:04:17.000Z",
  "title": "The stability index for dynamically defined Weierstrass functions",
  "authors": [
    "Charles P Walkden",
    "Tom Withers"
  ],
  "comment": "25 pages, 3 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $\\hat{T} : X \\times \\mathbb{R} \\to X \\times \\mathbb{R}$ given by $\\hat{T}(x,t) = (Tx, g_x(t))$ be a skew-product dynamical system where $T : X \\to X$ is a mixing conformal expanding map and, for each $x \\in X$, $g_x : \\mathbb{R} \\to \\mathbb{R}$ is an affine map of the form $g_x(t)=-f(x)+\\lambda(x)^{-1}t$. Under a suitable contraction hypotheses on $\\lambda$ there exists a measurable function $u: X \\to \\mathbb{R}$ such that $\\text{graph}(u) = \\{(x,u(x)) \\mid x\\in X\\}$ is $\\hat{T}$-invariant and divides $X \\times \\mathbb{R}$ into two regions, $\\mathbb{B}^+$ and $\\mathbb{B}^-$, consisting of points that are repelled under iteration by $\\hat{T}$ to $\\pm\\infty$. These two regions act as basins of attraction to $\\pm \\infty$ in the sense of Milnor. The two basins have a complicated local structure: a neighbourhood of a point $(x,t) \\in ^+$ will typically intersect $\\mathbb{B}^-$ in a set of positive measure. The stability index (as introduced by Podvigina and Ashwin \\cite{podviginaashwin:11} for general Milnor attractors) is the rate of polynomial decay of the measure of this intersection. We calculate the stability index at typical points in $X \\times \\mathbb{R}$. We also perform a multifractal analysis of the level sets of the stability index.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-07T21:04:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37D35",
      "37D45",
      "26A30"
    ],
    "keywords": [
      "stability index",
      "dynamically defined weierstrass functions",
      "general milnor attractors",
      "level sets",
      "mixing conformal expanding map"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}