C^0-rigidity of Poisson brackets

TL;DR

This paper proves that the maximum of the Poisson bracket of two functions is lower semi-continuous under the C^0 norm on symplectic manifolds, extending previous results and using Hofer metric geodesic theory.

Contribution

It establishes C^0-rigidity of the Poisson bracket functional for pairs of functions, extending prior semi-continuity results to a broader context.

Findings

Poisson bracket maximum is lower semi-continuous in C^0 topology

Extension of previous semi-continuity results by Cardin-Viterbo and Zapolsky

Failure of similar semi-continuity for multiple Poisson brackets with three or more functions

Abstract

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.