Constraints on symplectic quasi-states

TL;DR

This paper demonstrates that large measure portions of symplectic manifolds can be embedded into polydisks, and uses this to show certain constructions cannot produce nonlinear symplectic quasi-states in higher dimensions.

Contribution

It generalizes embedding results to symplectic manifolds and proves limitations on the construction of nonlinear symplectic quasi-states.

Findings

Large measure sets can be symplectically embedded into polydisks.

Certain soft constructions cannot produce nonlinear symplectic quasi-states in dimensions ≥4.

The embedding result constrains the form of symplectic quasi-states.

Abstract

We prove that given a closed connected symplectic manifold equipped with a Borel probability measure, an arbitrarily large portion of the measure can be covered by a symplectically embedded polydisk, generalizing a result of Schlenk. We apply this to constraints on symplectic quasi-states. Quasi-states are a certain class of not necessarily linear functionals on the algebra of continuous functions of a compact space. When the space is a symplectic manifold, a more restrictive subclass of symplectic quasi-states was introduced by Entov--Polterovich. We use our embedding result to prove that a certain `soft' construction of quasi-states, which is due to Aarnes, cannot yield nonlinear symplectic quasi-states in dimension at least four.