A New Approach to Numerical Computation of Hausdorff Dimension of Iterated Function Systems: Applications to Complex Continued Fractions

Abstract

In a previous paper, dealing with "Applications in $\mathbb{R}^1$," 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 is 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 IFS associated to complex continued fractions, (which may arise from an infinite iterated function system), we again approximate the eigenvalue problem by a collocation method, but now 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.