Instantaneous Hamiltonian displaceability and arbitrary symplectic squeezability for critically negligible sets

Yann Guggisberg, Fabian Ziltener
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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.
临界可忽略集合的瞬时哈密顿可置换性和任意交映可挤压性
如果一个度量空间的 $s$ 维 Hausdorffmeasure 消失,我们就称它为 $s$ 不可忽略空间。我们证明,$mathbb{R}^{2n}$的每一个可数$m$可校正子集都可以通过一个哈密顿衍射从每一个$(2n-m)$可忽略子集移出,而这个哈密顿衍射是任意地$C^\infty$接近于同一性的。因此,$\mathbb{R}^{2n}$的每一个可数$n$可校正且$n$不可忽略的子集都是任意可共挤的。这两个结果在$s$不可忽略假设中的参数$s$时都是尖锐的。我们的挤压结果的证明使用了折叠法。有可能,我们的折叠方法可以被修改以证明,对于开放单位球 $B^{2n}_1$ 的每一个可数 $(n-1)$ 直的封闭子集 $A$,$B^{2n}_1setminus A$ 的格罗莫夫宽度等于 $\pi$。这意味着 $A$ 不是一个障碍。
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