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$$\mathcal C^0$$-rigidity of characteristics in symplectic geometry. (English) Zbl 1186.53054
The main result (Theorem 1) says that a symplectic homeomorphism in the sense of Eliashberg and Gromov [e. g., Y. M. Eliashberg, Funct. Anal. Appl. 21, 227-232 (1987); translation from Funkts. Anal. Prilozh. 21, No. 3, 65–72 (1987; Zbl 0655.58015)] which preserves a smooth hypersurface also preserves its characteristic foliation. Let $$B$$ be a symplectic embedding of a Euclidean ball centered at origin into a symplectic manifold $$M$$ which sends $$\mathbb R^{2n-1}:= \mathbb R\times\mathbb C^{n-1}\subset \mathbb C^n$$ to a hyperspace $$S\subset M$$. $$B$$ is divided into two components $$S_+$$ and $$S_-$$. For a real number $$\varepsilon>0$$ and $$x, y\in S\cap B$$, the author uses a notion of continuous path of symplectomorphisms $$\Phi_t(t\in[0,1])$$ with common supports in $$B$$ for which there exists two open sets $$U_{\varepsilon}(x)$$ and $$U_{\varepsilon}(y)$$ contained in the $$\varepsilon$$-balls around $$x$$ and $$y$$ respectively, such that:
1. $$\Phi_0=Id$$,
2. $$\Phi_1(z)\in S_+$$ for all $$t\in ]0,1]$$ and $$z\in S\cap U_{\varepsilon}(x)$$,
3. $$\Phi_{1}(z)\in S_{-}$$ { for all} $$t\in ]0,1]$$ and $$z\in S\cap U_{\varepsilon}(y)$$,
4. $$\Phi_{t}(y)\in S$$ { for all} $$t\in [0,1]$$ and $$z\in S\backslash(U_{\varepsilon}(x)\cap U_{\varepsilon}(y))$$.
By using Gromov’s non-squeezing theorem it is shown (Proposition 1.3) that the path $$\Phi_t$$ exists if and only if $$x$$ and $$y$$ are on the same characteristic leaf. The author calls the path $$\Phi_t$$ an $$\varepsilon$$-symplectic hammer. By consecutive applications of $$\varepsilon$$-symplectic hammers, the main theorem is proved. The author adds a one-sided version (Theorem 1.2) on a smoothly bounded open set $$U\subset \mathbb R^4$$ symplectomorphic to $$B^4(1)$$ and its boundary $$\partial U$$.

##### MSC:
 53C24 Rigidity results 53D05 Symplectic manifolds (general theory) 57R17 Symplectic and contact topology in high or arbitrary dimension 53C12 Foliations (differential geometric aspects)
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