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$$C^0$$-rigidity of Poisson brackets. (English) Zbl 1197.53115
Fathi, Albert (ed.) et al., Symplectic topology and measure preserving dynamical systems. Papers of the AMS-IMS-SIAM joint summer research conference, Snowbird, UT, USA, July 1–5, 2007. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4892-0/pbk). Contemporary Mathematics 512, 25-32 (2010).
Summary: 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 the theory of geodesies of the Hofer metric on the group of Hamiltonian diffeomorphisms. We also discuss a failure of a similar semi-continuity phenomenon for iterated Poisson brackets of three or more functions.
For the entire collection see [Zbl 1186.57001].

##### MSC:
 53D35 Global theory of symplectic and contact manifolds 53D05 Symplectic manifolds (general theory)
##### Keywords:
symplectic manifold; Poisson bracket; $$C^0$$-norm; Hofer metric
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