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arXiv · 2604.20812

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

Jacob Brown
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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.

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Jacob Brown. 2026-04-22. Rigorous High-Order Hausdorff Dimension Estimation of Limit Sets of Continued Fraction Iterated Function Systems via B-Splines. https://arxiv.org/abs/2604.20812

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