Separation of horocycle orbits on moduli space in genus 2
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910505506963456 |
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| author | Rached, John |
| author_facet | Rached, John |
| contents | We prove a quantitative closing lemma for the horocycle flow induced by the $\mathrm{SL}(2,\mathbb{R})$-action on the moduli space of Abelian differentials with a double-order zero on surfaces of genus 2. The proof proceeds via construction of a Margulis function measuring the discretized fractal dimension of separation of a horocycle orbit of a point from itself, in a direction transverse to the $\mathrm{SL}(2,\mathbb{R})$-orbit. From this, we deduce that small transversal separation guarantees the existence of a nearby point with a pseudo-Anosov in its Veech group. This is reminiscent of the initial dimension phases in Bourgain-Gamburd for random walks on compact groups, Bourgain-Lindenstrauss-Furman-Mozes for quantitative equidistribution in tori, and quantitative equidistribution of horocycle flow for a product of $\mathrm{SL}(2,\mathbb{R})$ with itself due to Lindenstrauss-Mohammadi-Wang, and multiple other works. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_19527 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Separation of horocycle orbits on moduli space in genus 2 Rached, John Dynamical Systems General Topology We prove a quantitative closing lemma for the horocycle flow induced by the $\mathrm{SL}(2,\mathbb{R})$-action on the moduli space of Abelian differentials with a double-order zero on surfaces of genus 2. The proof proceeds via construction of a Margulis function measuring the discretized fractal dimension of separation of a horocycle orbit of a point from itself, in a direction transverse to the $\mathrm{SL}(2,\mathbb{R})$-orbit. From this, we deduce that small transversal separation guarantees the existence of a nearby point with a pseudo-Anosov in its Veech group. This is reminiscent of the initial dimension phases in Bourgain-Gamburd for random walks on compact groups, Bourgain-Lindenstrauss-Furman-Mozes for quantitative equidistribution in tori, and quantitative equidistribution of horocycle flow for a product of $\mathrm{SL}(2,\mathbb{R})$ with itself due to Lindenstrauss-Mohammadi-Wang, and multiple other works. |
| title | Separation of horocycle orbits on moduli space in genus 2 |
| topic | Dynamical Systems General Topology |
| url | https://arxiv.org/abs/2406.19527 |