CmC^m Eigenfunctions of Perron–Frobenius operators and a new approach to numerical computation of Hausdorff dimension: applications in R1\mathbb R^1

  • Richard S. Falk

    Rutgers University, Piscataway, USA
  • Roger D. Nussbaum

    Rutgers University, Piscataway, USA
$C^m$ Eigenfunctions of Perron–Frobenius operators and a new approach to numerical computation of Hausdorff dimension: applications in $\mathbb R^1$ cover
Download PDF

A subscription is required to access this article.

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 that we consider here, our methods require only C3C^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 LsL_s. The operators LsL_s can typically be studied in many different Banach spaces. Here, unlike most of the literature, we study LsL_s in a Banach space of real-valued, CkC^k functions, k2k \ge 2. We note that LsL_s is not compact, but has essential spectral radius ρs\rho_s strictly less than the spectral radius λs\lambda_s and possesses a strictly positive CkC^k eigenfunction vsv_s with eigenvalue λs\lambda_s. Under appropriate assumptions on the IFS, the Hausdorff dimension of the invariant set of the IFS is the value s=ss=s_* for which λs=1\lambda_s =1. This eigenvalue problem is then approximated by a collocation method using continuous piecewise linear functions. Using the theory of positive linear operators and explicit a priori bounds on the derivatives of the strictly positive eigenfunction vsv_s, we give rigorous upper and lower bounds for the Hausdorff dimension ss_*, and these bounds converge to ss_* as the mesh size approaches zero.

Cite this article

Richard S. Falk, Roger D. Nussbaum, CmC^m Eigenfunctions of Perron–Frobenius operators and a new approach to numerical computation of Hausdorff dimension: applications in R1\mathbb R^1. J. Fractal Geom. 5 (2018), no. 3, pp. 279–337

DOI 10.4171/JFG/62