Local middle dimensional symplectic non-squeezing in the analytic setting

TL;DR

This paper establishes a middle-dimensional non-squeezing property for analytic symplectic embeddings, showing volume preservation in projections onto symplectic subspaces under certain conditions.

Contribution

It proves a new middle-dimensional non-squeezing theorem for analytic symplectic embeddings, extending classical results to a broader setting.

Findings

Existence of a positive radius function ensuring volume inequalities

Volume preservation in projections onto symplectic subspaces

Application of analytic methods to symplectic non-squeezing

Abstract

We prove the following middle-dimensional non-squeezing result for analytic symplectic embeddings of domains in $\mathbb{R}^{2n}$. Let $\varphi: D \hookrightarrow \mathbb{R}^{2n}$ be an analytic symplectic embedding of a domain $D \subset \mathbb{R}^{2n}$ and $P$ be a symplectic projector onto a linear $2k$-dimensional symplectic subspace $V\subset \mathbb{R}^{2n}$. Then there exists a positive function $r_0:D\rightarrow (0,+ \infty)$, bounded away from $0$ on compact subsets $K \subset D$, such that the inequality $Vol_{2k}(P\varphi (B_r(x)),\omega ^k _{0|V})\geq \pi^{k} r^{2k}$ holds for every $x \in D$ and for every $r < r_0(x)$. This claim will be deduced from an analytic middle-dimensional non-squeezing result (stated by considering paths of symplectic embeddings) whose proof will be carried on by taking advantage of a work by \'{A}lvarez Paiva and Balacheff.