On the graph of the dimension function of the Lagrange and Markov spectra
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| Format: | Preprint |
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2022
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| _version_ | 1866915951153250304 |
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| author | Matheus, Carlos Moreira, Carlos Gustavo Vytnova, Polina |
| author_facet | Matheus, Carlos Moreira, Carlos Gustavo Vytnova, Polina |
| contents | We study the graph of the function $d(t)$ encoding the Hausdorff dimensions of the classical Lagrange and Markov spectra with half-infinite lines of the form $(-\infty, t)$. For this sake, we use the fact that the Hausdorff dimension of dynamically Cantor sets drop after erasing an element of its Markov partition to determine twelve nontrivial plateaux of $d(t)$. Next, we employ rigorous numerical methods (from our recent joint paper with Pollicott) to produce approximations of the graph of $d(t)$ between these twelve plateaux. As a corollary, we prove that the largest ten non-trivial plateaux of $d(t)$ are exactly those plateaux with lengths $> 0.005$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2212_11371 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the graph of the dimension function of the Lagrange and Markov spectra Matheus, Carlos Moreira, Carlos Gustavo Vytnova, Polina Number Theory Dynamical Systems 11A55, 11J06, 37D35, 37M25 We study the graph of the function $d(t)$ encoding the Hausdorff dimensions of the classical Lagrange and Markov spectra with half-infinite lines of the form $(-\infty, t)$. For this sake, we use the fact that the Hausdorff dimension of dynamically Cantor sets drop after erasing an element of its Markov partition to determine twelve nontrivial plateaux of $d(t)$. Next, we employ rigorous numerical methods (from our recent joint paper with Pollicott) to produce approximations of the graph of $d(t)$ between these twelve plateaux. As a corollary, we prove that the largest ten non-trivial plateaux of $d(t)$ are exactly those plateaux with lengths $> 0.005$. |
| title | On the graph of the dimension function of the Lagrange and Markov spectra |
| topic | Number Theory Dynamical Systems 11A55, 11J06, 37D35, 37M25 |
| url | https://arxiv.org/abs/2212.11371 |