Rigorous High-Order Hausdorff Dimension Estimation of Limit Sets of Continued Fraction Iterated Function Systems via B-Splines

TL;DR

This paper introduces a rigorous numerical method using B-splines to estimate the Hausdorff dimensions of limit sets from continued fraction iterated function systems, achieving higher-order convergence and computational flexibility.

Contribution

The authors develop a novel B-spline based finite element approach for precise Hausdorff dimension estimation, extending Falk and Nussbaum's positivity results to this context.

Findings

The method provides rigorous upper and lower bounds for Hausdorff dimensions.

Numerical results confirm higher-order convergence in one and two dimensions.

The approach demonstrates computational advantages over traditional methods.

Abstract

We develop a method for the rigorous estimation of Hausdorff dimensions of limit sets produced by continued fraction iterated function systems. Our method is based on the approximation of a Perron-Frobenius operator using the finite element method with B-splines as the choice of basis functions. This choice provides key numerical advantages including higher-order convergence and computational flexibility. We prove an analogue of Falk and Nussbaum's result on "hidden positivity" for B-spline quasi-interpolants to give rigorous upper and lower bounds for the Hausdorff dimensions of various limit sets. We provide numerical results to verify both the rigor and higher-order convergence of our method for quadratic B-spline interpolants in one and two dimensions.