C^0-rigidity of Poisson brackets

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Consider a functional associating to a pair of compactly supported smooth functions on a symplectic manifold the maximum of their Poisson bracket. We show that this functional is lower semi-continuous with respect to the product uniform (C^0) norm on the space of pairs of such functions. This extends previous results of Cardin-Viterbo and Zapolsky. The proof involves theory of geodesics of the Hofer metric on the group of Hamiltonian diffeomorphisms. We also discuss a failure of a similar semi-continuity phenomenon for multiple Poisson brackets of three or more functions.
Latex, 11 pages

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Symplectic Geometry, Classical Analysis and ODEs, 53D35, 53D05

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http://salesiana.dossiersoluciones.com/handle/123456789/144458

Consulte el texto completo en el siguiente enlace:

https://arxiv.org/abs/0712.2913
 http://arxiv.org/abs/0712.2913

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arXiv