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