{"title":"Instantaneous Hamiltonian displaceability and arbitrary symplectic squeezability for critically negligible sets","authors":"Yann Guggisberg, Fabian Ziltener","doi":"arxiv-2408.17444","DOIUrl":null,"url":null,"abstract":"We call a metric space $s$-negligible iff its $s$-dimensional Hausdorff\nmeasure vanishes. We show that every countably $m$-rectifiable subset of\n$\\mathbb{R}^{2n}$ can be displaced from every $(2n-m)$-negligible subset by a\nHamiltonian diffeomorphism that is arbitrarily $C^\\infty$-close to the\nidentity. As a consequence, every countably $n$-rectifiable and $n$-negligible\nsubset of $\\mathbb{R}^{2n}$ is arbitrarily symplectically squeezable. Both\nresults are sharp w.r.t. the parameter $s$ in the $s$-negligibility assumption. The proof of our squeezing result uses folding. Potentially, our folding\nmethod can be modified to show that the Gromov width of $B^{2n}_1\\setminus A$\nequals $\\pi$ for every countably $(n-1)$-rectifiable closed subset $A$ of the\nopen unit ball $B^{2n}_1$. This means that $A$ is not a barrier.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Symplectic Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.17444","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 call a metric space $s$-negligible iff its $s$-dimensional Hausdorff
measure vanishes. We show that every countably $m$-rectifiable subset of
$\mathbb{R}^{2n}$ can be displaced from every $(2n-m)$-negligible subset by a
Hamiltonian diffeomorphism that is arbitrarily $C^\infty$-close to the
identity. As a consequence, every countably $n$-rectifiable and $n$-negligible
subset of $\mathbb{R}^{2n}$ is arbitrarily symplectically squeezable. Both
results are sharp w.r.t. the parameter $s$ in the $s$-negligibility assumption. The proof of our squeezing result uses folding. Potentially, our folding
method can be modified to show that the Gromov width of $B^{2n}_1\setminus A$
equals $\pi$ for every countably $(n-1)$-rectifiable closed subset $A$ of the
open unit ball $B^{2n}_1$. This means that $A$ is not a barrier.