Rigorous High-Order Hausdorff Dimension Estimation of Limit Sets of Continued Fraction Iterated Function Systems via B-Splines
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866913055696224256 |
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| author | Brown, Jacob |
| author_facet | Brown, Jacob |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_20812 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Rigorous High-Order Hausdorff Dimension Estimation of Limit Sets of Continued Fraction Iterated Function Systems via B-Splines Brown, Jacob Numerical Analysis 65J10 (Primary) 65D07 (Secondary) 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. |
| title | Rigorous High-Order Hausdorff Dimension Estimation of Limit Sets of Continued Fraction Iterated Function Systems via B-Splines |
| topic | Numerical Analysis 65J10 (Primary) 65D07 (Secondary) |
| url | https://arxiv.org/abs/2604.20812 |