Local non-squeezing theorems and stability.

*(English)*Zbl 0837.58014A fundamental theorem in symplectic topology is the non-squeezing theorem, first proved by Gromov for a range of manifolds including Euclidean space, and later generalized to all manifolds by the authors. In this paper, local versions of this theorem are considered.

Suppose \((W, \Omega)= (M\times D, \omega\oplus \sigma)\) is a symplectic cylinder, where \(D\) is a closed 2-disc of \(\sigma\)-area \(\pi\) and \((M, \omega)\) is some symplectic manifold. If \(S\) is a compact subset of \(W\) whose boundary is a smooth hypersurface, one can ask when \(S\) can be moved symplectically to lie strictly inside \(W\). The authors’ main theorem gives an essentially complete answer to this question. The crux of the problem lies in the geometry of \(S\) near the points which meet the boundary \(\partial W\) of the cylinder, and the most interesting case occurs when \(S\) meets \(\partial W\) along some closed characteristic \(\chi\times \partial D\). In this case, the problem is closely connected to the properties of the linearization of the characteristic flow around this closed orbit.

Suppose \((W, \Omega)= (M\times D, \omega\oplus \sigma)\) is a symplectic cylinder, where \(D\) is a closed 2-disc of \(\sigma\)-area \(\pi\) and \((M, \omega)\) is some symplectic manifold. If \(S\) is a compact subset of \(W\) whose boundary is a smooth hypersurface, one can ask when \(S\) can be moved symplectically to lie strictly inside \(W\). The authors’ main theorem gives an essentially complete answer to this question. The crux of the problem lies in the geometry of \(S\) near the points which meet the boundary \(\partial W\) of the cylinder, and the most interesting case occurs when \(S\) meets \(\partial W\) along some closed characteristic \(\chi\times \partial D\). In this case, the problem is closely connected to the properties of the linearization of the characteristic flow around this closed orbit.

Reviewer: W.J.Satzer jun.(St.Paul)

##### MSC:

37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |

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