Local middle dimensional symplectic non-squeezing in the analytic setting

Abstract

We prove the following middle-dimensional non-squeezing result for analytic symplectic embeddings of domains in ${\mathbb{R}}^{2n}$. Let $\phi :D\hookrightarrow {\mathbb{R}}^{2n}$ 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 $r_0:D\to (0,+\infty )$, bounded away from $0$ on compact subsets $K\subset D$, such that the inequality $Vo{l}_{2k}(P\phi (B_r(x)),{\omega }_{0|V}^k)\ge {\pi }^k{r}^{2k}$ 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 Álvarez Paiva and Balacheff.

27 pages

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