Extreme value theorem for geodesic flow on the quotient of the theta group
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866914378694000640 |
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| author | Kim, Jaelin Lee, Seul Bee Lim, Seonhee |
| author_facet | Kim, Jaelin Lee, Seul Bee Lim, Seonhee |
| contents | We establish an extreme value theorem for the geodesic flow on the hyperbolic surface $Θ\backslash\mathbb{H}^2$ associated with the theta group $Θ$. To capture excursions into both cusps of this surface, we introduce a generalized continued fraction algorithm obtained by splicing the even and odd-odd continued fraction maps into a single dynamical system. We prove that the natural extension of this map is isomorphic to the first return map of the geodesic flow on a suitable cross section. Using spectral properties of the associated transfer operator, we derive a Galambos-type extreme value law for the digits of the spliced continued fraction. This symbolic result is then translated into a geometric extreme value theorem describing maximal cusp excursions of geodesics on $Θ\backslash\mathbb{H}^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_07649 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Extreme value theorem for geodesic flow on the quotient of the theta group Kim, Jaelin Lee, Seul Bee Lim, Seonhee Dynamical Systems Number Theory 37D40, 11J70, 11K50 We establish an extreme value theorem for the geodesic flow on the hyperbolic surface $Θ\backslash\mathbb{H}^2$ associated with the theta group $Θ$. To capture excursions into both cusps of this surface, we introduce a generalized continued fraction algorithm obtained by splicing the even and odd-odd continued fraction maps into a single dynamical system. We prove that the natural extension of this map is isomorphic to the first return map of the geodesic flow on a suitable cross section. Using spectral properties of the associated transfer operator, we derive a Galambos-type extreme value law for the digits of the spliced continued fraction. This symbolic result is then translated into a geometric extreme value theorem describing maximal cusp excursions of geodesics on $Θ\backslash\mathbb{H}^2$. |
| title | Extreme value theorem for geodesic flow on the quotient of the theta group |
| topic | Dynamical Systems Number Theory 37D40, 11J70, 11K50 |
| url | https://arxiv.org/abs/2603.07649 |