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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.06330 |
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| _version_ | 1866908343392534528 |
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| author | Chousionis, Vasileios Leykekhman, Dmitriy Urbański, Mariusz Wendt, Erik |
| author_facet | Chousionis, Vasileios Leykekhman, Dmitriy Urbański, Mariusz Wendt, Erik |
| contents | We develop a versatile framework which allows us to rigorously estimate the Hausdorff dimension of maximal conformal graph directed Markov systems in $\mathbb{R}^n$ for $n \geq 2$. Our method is based on piecewise linear approximations of the eigenfunctions of the Perron-Frobenius operator via a finite element framework for discretization and iterative mesh schemes. One key element in our approach is obtaining bounds for the derivatives of these eigenfunctions, which, besides being essential for the implementation of our method, are of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_06330 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rigorous Hausdorff dimension estimates for conformal fractals Chousionis, Vasileios Leykekhman, Dmitriy Urbański, Mariusz Wendt, Erik Dynamical Systems Numerical Analysis 37D35, 37C30, 28A80, 65J10, 11J70 We develop a versatile framework which allows us to rigorously estimate the Hausdorff dimension of maximal conformal graph directed Markov systems in $\mathbb{R}^n$ for $n \geq 2$. Our method is based on piecewise linear approximations of the eigenfunctions of the Perron-Frobenius operator via a finite element framework for discretization and iterative mesh schemes. One key element in our approach is obtaining bounds for the derivatives of these eigenfunctions, which, besides being essential for the implementation of our method, are of independent interest. |
| title | Rigorous Hausdorff dimension estimates for conformal fractals |
| topic | Dynamical Systems Numerical Analysis 37D35, 37C30, 28A80, 65J10, 11J70 |
| url | https://arxiv.org/abs/2408.06330 |