Hidden positivity and a new approach to numerical computation of Hausdorff dimension: higher order methods

  • Richard S. Falk

    Rutgers University, Piscataway, USA
  • Roger D. Nussbaum

    Rutgers University, Piscataway, USA
Hidden positivity and a new approach to numerical computation of Hausdorff dimension: higher order methods cover
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Abstract

In 2018, the authors developed a new approach to the computation of the Hausdorff dimension of the invariant set of an iterated function system or IFS. In this paper, we extend this approach to incorporate high order approximation methods. We again rely on the fact that we can associate to the IFS a parametrized family of positive, linear, Perron–Frobenius operators LsL_s, an idea known in varying degrees of generality for many years. Although LsL_s is not compact in the setting we consider, it possesses a strictly positive CmC^m eigenfunction vsv_s with eigenvalue R(Ls)R(L_s) for arbitrary mm and all other points zz in the spectrum of LsL_s satisfy zb|z| \le b for some constant b<R(Ls)b < R(L_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 R(Ls)=1R(L_s) =1. This eigenvalue problem is then approximated by a collocation method at the extended Chebyshev points of each subinterval using continuous piecewise polynomials of arbitrary degree rr. Using an extension of the Perron theory of positive matrices to matrices that map a cone KK to its interior 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 rapidly to ss_* as the mesh size decreases and/or the polynomial degree increases.

Cite this article

Richard S. Falk, Roger D. Nussbaum, Hidden positivity and a new approach to numerical computation of Hausdorff dimension: higher order methods. J. Fractal Geom. 9 (2022), no. 1/2, pp. 23–72

DOI 10.4171/JFG/111