C^0-rigidity of Poisson brackets

Details C^0-rigidity of Poisson brackets C^0-rigidity of Poisson brackets

2007-12-18

arxiv.org/abs/0712.2913v1

Mathematics

Symplectic Geometry

Latex, 11 pages

Scientific paper

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.

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