Inducing recurrent flows by twisting on infinite surfaces with unbounded cuffs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909347616915456 |
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| author | Hakobyan, Hrant Pandazis, Michael Saric, Dragomir |
| author_facet | Hakobyan, Hrant Pandazis, Michael Saric, Dragomir |
| contents | A Riemann surface $X$ is parabolic if and only if the geodesic flow (for the hyperbolic metric) on the unit tangent bundle of $X$ is ergodic. Consider a Riemann surface $X$ with a single topological end and a sequence $α_n$ of pairwise disjoint, simple closed geodesics converging to the end, called {\it cuffs}. Basmajian, the first and the third author, proved that when the lengths $\ell (α_n)$ of cuffs are at most $2\log n$, the surface $X$ is parabolic. One could expect that having arbitrary large cuff lengths $\ell (α_n)$ (think of $\ell (α_n)=n!^{n!}$) would allow the geodesic flow to escape to infinity, thus making $X$ not parabolic.
Contrary to this and motivated by their proof of the Surface Subgroup Theorem, Kahn and Marković conjectured that for every choice of lengths $\ell (α_n)$, there is a choice of twists that would make $X$ parabolic. We show that their conjecture is essentially true. Namely, for any sequence of positive numbers $\{ a_n\}$, there is a choice of lengths $\ell (α_n)\geq a_n$ such that the (relative) twists by $1/2$ make $X$ parabolic. This result extends to the surfaces with countably many ends while it does not hold for uncountably many ends. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_10057 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Inducing recurrent flows by twisting on infinite surfaces with unbounded cuffs Hakobyan, Hrant Pandazis, Michael Saric, Dragomir Dynamical Systems Complex Variables Geometric Topology 30F15, 30F25, 37A25 A Riemann surface $X$ is parabolic if and only if the geodesic flow (for the hyperbolic metric) on the unit tangent bundle of $X$ is ergodic. Consider a Riemann surface $X$ with a single topological end and a sequence $α_n$ of pairwise disjoint, simple closed geodesics converging to the end, called {\it cuffs}. Basmajian, the first and the third author, proved that when the lengths $\ell (α_n)$ of cuffs are at most $2\log n$, the surface $X$ is parabolic. One could expect that having arbitrary large cuff lengths $\ell (α_n)$ (think of $\ell (α_n)=n!^{n!}$) would allow the geodesic flow to escape to infinity, thus making $X$ not parabolic. Contrary to this and motivated by their proof of the Surface Subgroup Theorem, Kahn and Marković conjectured that for every choice of lengths $\ell (α_n)$, there is a choice of twists that would make $X$ parabolic. We show that their conjecture is essentially true. Namely, for any sequence of positive numbers $\{ a_n\}$, there is a choice of lengths $\ell (α_n)\geq a_n$ such that the (relative) twists by $1/2$ make $X$ parabolic. This result extends to the surfaces with countably many ends while it does not hold for uncountably many ends. |
| title | Inducing recurrent flows by twisting on infinite surfaces with unbounded cuffs |
| topic | Dynamical Systems Complex Variables Geometric Topology 30F15, 30F25, 37A25 |
| url | https://arxiv.org/abs/2410.10057 |