Title:         Hidden Positivity and a New Approach to Numerical Computation of
               Hausdorff Dimension: Higher Order Methods

Authors:       Richard S. Falk and Roger D. Nussbaum

Source:        J. Fractal Geomety vol. 9 (2022) 23-72.

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 $L_s$, an
  idea known in varying degrees of generality for many years. Although
  $L_s$ is not compact in the setting we consider, it possesses a
  strictly positive $C^m$ eigenfunction $v_s$ with eigenvalue
  $R(L_s)$ for arbitrary $m$ and all other points $z$ in the
  spectrum of $L_s$ satisfy $|z| \le b$ for some constant $b <
  R(L_s)$. Under appropriate assumptions on the IFS, the Hausdorff
  dimension of the invariant set of the IFS is the value $s=s_*$ for
  which $R(L_s) =1$.  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 $r$.  Using an extension of the Perron theory of
  positive matrices to matrices that map a cone $K$ to its interior
  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
  rapidly to $s_*$ as the mesh size decreases and/or the polynomial
  degree increases.

Key words:  Hausdorff dimension, positive transfer operators, 
            continued fractions   

AMS(MOS) subject classifications. Primary 11K55, 37C30; Secondary: 65J10