Local middle dimensional symplectic non-squeezing in the analytic setting

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 $r_0 : D \to (0,+ \infty)$, bounded away from $0$ on compact subsets $K \subset D$, such that the inequality $V ol_{2k} (P \varphi (B_r (x)), \omega^k_{0 \vert V}) \geq \pi^k r^{2k}$ holds for every $x \in D$ and for every $r \lt r_0 (x)$. 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.

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This work is partially supported by the DFG grant AB 360/1-1.

Received 22 March 2016

Accepted 11 July 2018

Published 26 July 2019