{
  "id": "1601.06737",
  "version": "v1",
  "published": "2016-01-25T19:32:49.000Z",
  "updated": "2016-01-25T19:32:49.000Z",
  "title": "C^m Eigenfunctions of Perron-Frobenius Operators and a New Approach to Numerical Computation of Hausdorff Dimension",
  "authors": [
    "Richard S. Falk",
    "Roger D. Nussbaum"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We develop a new approach to the computation of the Hausdorff dimension of the invariant set of an iterated function system or IFS. In the one dimensional case, our methods require only C^3 regularity of the maps in the IFS. The key idea, which has been known in varying degrees of generality for many years, is to associate to the IFS a parametrized family of positive, linear, Perron-Frobenius operators L_s. The operators L_s can typically be studied in many different Banach spaces. Here, unlike most of the literature, we study L_s in a Banach space of real-valued, C^k functions, k >= 2; and we note that L_s is not compact, but has a strictly positive eigenfunction v_s with positive eigenvalue lambda_s equal to the spectral radius of L_s. Under appropriate assumptions on the IFS, the Hausdorff dimension of the invariant set of the IFS is the value s=s_* for which lambda_s =1. This eigenvalue problem is then approximated by a collocation method using continuous piecewise linear functions (in one dimension) or bilinear functions (in two dimensions). Using the theory of positive linear operators and explicit a priori bounds on the derivatives of the strictly positive eigenfunction v_s, we give rigorous upper and lower bounds for the Hausdorff dimension s_*, and these bounds converge to s_* as the mesh size approaches zero.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-25T19:32:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K55",
      "37C30",
      "65D05"
    ],
    "keywords": [
      "hausdorff dimension",
      "perron-frobenius operators",
      "numerical computation",
      "banach space",
      "strictly positive eigenfunction"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}