On an extreme value law for the unipotent flow on $\mathrm{SL}_2(\mathbb{R})/\mathrm{SL}_2(\mathbb{Z})$
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866914911557255168 |
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| author | Kirsebom, Maxim Mallahi-Karai, Keivan |
| author_facet | Kirsebom, Maxim Mallahi-Karai, Keivan |
| contents | We study an extreme value distribution for the unipotent flow on the modular surface $\mathrm{SL}_2(\mathbb{R})/\mathrm{SL}_2(\mathbb{Z})$. Using tools from homogenous dynamics and geometry of numbers we prove the existence of a continuous distribution function $F(r)$ for the normalized deepest cusp excursions of the unipotent flow. We find closed analytic formulas for $F(r)$ for $r \in [-\frac{1}{2} \log 2, \infty)$, and establish asymptotic behavior of $F(r)$ as $r \to -\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_07283 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On an extreme value law for the unipotent flow on $\mathrm{SL}_2(\mathbb{R})/\mathrm{SL}_2(\mathbb{Z})$ Kirsebom, Maxim Mallahi-Karai, Keivan Dynamical Systems Number Theory Probability 37A17, 60G70, 11H06 We study an extreme value distribution for the unipotent flow on the modular surface $\mathrm{SL}_2(\mathbb{R})/\mathrm{SL}_2(\mathbb{Z})$. Using tools from homogenous dynamics and geometry of numbers we prove the existence of a continuous distribution function $F(r)$ for the normalized deepest cusp excursions of the unipotent flow. We find closed analytic formulas for $F(r)$ for $r \in [-\frac{1}{2} \log 2, \infty)$, and establish asymptotic behavior of $F(r)$ as $r \to -\infty$. |
| title | On an extreme value law for the unipotent flow on $\mathrm{SL}_2(\mathbb{R})/\mathrm{SL}_2(\mathbb{Z})$ |
| topic | Dynamical Systems Number Theory Probability 37A17, 60G70, 11H06 |
| url | https://arxiv.org/abs/2209.07283 |