$C^m$ Eigenfunctions of Perron-Frobenius Operators and a New Approach to Numerical Computation of Hausdorff Dimension: Applications in $\mathbb{R}^1$

TL;DR

This paper introduces a novel numerical method for calculating the Hausdorff dimension of invariant sets in one-dimensional iterated function systems, leveraging $C^m$ eigenfunctions of Perron-Frobenius operators.

Contribution

It develops a new approach using $C^k$ function spaces and eigenfunction analysis to accurately approximate Hausdorff dimensions with rigorous bounds.

Findings

The method provides converging bounds for Hausdorff dimension estimates.

Eigenfunctions of Perron-Frobenius operators can be effectively approximated in $C^k$ spaces.

The approach requires only $C^3$ regularity of the IFS maps.

Abstract

We develop a new approach to the computation of the Hausdorff dimension of the invariant set of an iterated function system or IFS. In the one dimensional case that we consider here, our methods require only $C^3$ regularity of the maps in the IFS. 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 $L_s$. The operators $L_s$ can typically be studied in many different Banach spaces. Here, unlike most of the literature, we study $L_s$ in a Banach space of real-valued, $C^k$ functions, $k \ge 2$. We note that $L_s$ is not compact, but has essential spectral radius $\rho_s$ strictly less than the spectral radius $\lambda_s$ and possesses a strictly positive $C^k$ eigenfunction $v_s$ with eigenvalue $\lambda_s$. Under appropriate assumptions on the IFS, the Hausdorff dimension of the invariant set of the IFS is the value $s=s_*$ for which $\lambda_s =1$. This eigenvalue problem is then approximated by a collocation method using continuous piecewise linear functions. Using the theory of positive linear operators 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 to $s_*$ as the mesh size approaches zero.