Poisson brackets and symplectic invariants

TL;DR

This paper introduces new symplectic invariants derived from Poisson brackets, utilizing Floer theory, with applications in approximation theory and Hamiltonian dynamics.

Contribution

It presents novel invariants for collections of compact subsets in symplectic manifolds, connecting Poisson brackets with Floer theory for the first time.

Findings

Invariants are non-trivial and detect symplectic properties.

Applications include approximation theory on symplectic manifolds.

Results impact Hamiltonian dynamics analysis.

Abstract

We introduce new invariants associated to collections of compact subsets of a symplectic manifold. They are defined through an elementary-looking variational problem involving Poisson brackets. The proof of the non-triviality of these invariants involves various flavors of Floer theory. We present applications to approximation theory on symplectic manifolds and to Hamiltonian dynamics.