{"title":"Separation of horocycle orbits on moduli space in genus 2","authors":"John Rached","doi":"arxiv-2406.19527","DOIUrl":null,"url":null,"abstract":"We prove a quantitative closing lemma for the horocycle flow induced by the\n$\\mathrm{SL}(2,\\mathbb{R})$-action on the moduli space of Abelian differentials\nwith a double-order zero on surfaces of genus 2. The proof proceeds via\nconstruction of a Margulis function measuring the discretized fractal dimension\nof separation of a horocycle orbit of a point from itself, in a direction\ntransverse to the $\\mathrm{SL}(2,\\mathbb{R})$-orbit. From this, we deduce that\nsmall transversal separation guarantees the existence of a nearby point with a\npseudo-Anosov in its Veech group. This is reminiscent of the initial dimension\nphases in Bourgain-Gamburd for random walks on compact groups,\nBourgain-Lindenstrauss-Furman-Mozes for quantitative equidistribution in tori,\nand quantitative equidistribution of horocycle flow for a product of\n$\\mathrm{SL}(2,\\mathbb{R})$ with itself due to Lindenstrauss-Mohammadi-Wang,\nand multiple other works.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"1 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - General Topology","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2406.19527","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
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.