Title: A New Approach to Numerical Computation of Hausdorff Dimension
of Iterated Function Systems: Applications to Complex Continued
Fractions
Authors: Richard S. Falk and Roger D. Nussbaum
Source: Integral Equations Operator Theory 90 (2018), no. 5, 90:61.
Abstract:
In a previous paper \cite{hdcomp1}, the authors developed a new approach to
the computation of the Hausdorff dimension of the invariant set of an
iterated function system or IFS and studied some applications in one
dimension. 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 $L_s$. In our context, $L_s$
is studied in a space of $C^m$ functions and is not compact. Nevertheless,
it has a strictly positive $C^m$ 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 $\lambda_s =1$. To compute
the Hausdorff dimension of an invariant set for an IFS associated to complex
continued fractions, (which may arise from an infinite iterated function
system), we approximate the eigenvalue problem by a collocation method using
continuous piecewise bilinear functions. Using the theory of positive
linear operators and explicit a priori bounds on the partial derivatives of
the strictly positive eigenfunction $v_s$, we are able to give rigorous
upper and lower bounds for the Hausdorff dimension $s_*$, and these bounds
converge to $s_*$ as the mesh size approaches zero. We also demonstrate by
numerical computations that improved estimates can be obtained by the use of
higher order piecewise tensor product polynomial approximations, although
the present theory does not guarantee that these are strict upper and lower
bounds. An important feature of our approach is that it also applies to the
much more general problem of computing approximations to the spectral radius
of positive transfer operators, which arise in many other applications.
Key words: Hausdorff dimension, positive transfer operators,
continued fractions
AMS(MOS) subject classifications. Primary 11K55, 37C30; Secondary: 65D05