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Constraints on symplectic quasi-states

Adi Dickstein, Frol Zapolsky

TL;DR

The paper shows that on a closed connected symplectic manifold $(M,\omega)$ with $\dim M\ge 4$ and $H^1(M;\mathbb{Z})=0$, any symplectic Aarnes quasi-state $\zeta$ must be a delta-measure, ruling out nonlinear Aarnes-type quasi-states in this setting. The authors achieve this by constructing, for any $\varepsilon>0$, a symplectic embedding of a polydisk that covers most of a given probability measure $\mu$, and then using involutive maps to pushforward $\zeta$ to a linear quasi-state; the Aarnes representation then forces a delta-measure. A key step is identifying a $\,\zeta$-superheavy Lagrangian torus with accumulated mass $\ge 1/2$, which via a perturbation argument yields a point mass. Consequently, soft Aarnes-type constructions cannot yield nonlinear symplectic quasi-states in dimension $\ge 4$, highlighting a fundamental distinction between symplectic and volume-geometric structures and guiding expectations for nonlinear symplectic quasi-states in higher dimensions.

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.

Constraints on symplectic quasi-states

TL;DR

The paper shows that on a closed connected symplectic manifold with and , any symplectic Aarnes quasi-state must be a delta-measure, ruling out nonlinear Aarnes-type quasi-states in this setting. The authors achieve this by constructing, for any , a symplectic embedding of a polydisk that covers most of a given probability measure , and then using involutive maps to pushforward to a linear quasi-state; the Aarnes representation then forces a delta-measure. A key step is identifying a -superheavy Lagrangian torus with accumulated mass , which via a perturbation argument yields a point mass. Consequently, soft Aarnes-type constructions cannot yield nonlinear symplectic quasi-states in dimension , highlighting a fundamental distinction between symplectic and volume-geometric structures and guiding expectations for nonlinear symplectic quasi-states in higher dimensions.

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.
Paper Structure (18 sections, 30 theorems, 87 equations, 14 figures)

This paper contains 18 sections, 30 theorems, 87 equations, 14 figures.

Table of Contents

  1. Introduction and results
  2. Quasi-states and the main result
  3. Symplectic embeddings and an overview of the proof
  4. Discussion
  5. Preliminaries
  6. Measure theory
  7. Symplectic geometry
  8. Quasi-states and topological measures
  9. Constructing the embedding
  10. Embedding cubes
  11. Embedding polydisks into cubes
  12. Preparing the base
  13. The lift
  14. The folding
  15. Covering most of the measure by a polydisk
  16. ...and 3 more sections

Key Result

Theorem 1.2

Let $X$ be a connected finite CW-complex with $H^1(X;{\mathbb{Z}}) = 0$, and let $\mu$ be a Borel probability measure on $X$ with the property that there do not exist solid closed disjoint sets $C,C'\subset X$, such that $\mu(C)=\mu(C')=\frac{1}{2}$. Then there exists a unique quasi-state $\zeta$ on

Figures (14)

  • Figure 1: The lattice $\Sigma(y,a)$ in dimension $2$
  • Figure 2: The decomposition $M=C\sqcup\bigsqcup_iU_i$, where $C$ is the union of the curves, while the $U_i$ are the components of the complement.
  • Figure 3: The graph of the function $h$ (Section \ref{['sss:prep_base']})
  • Figure 4: The image of $\Theta$
  • Figure 5: One choice of the enumeration $e$
  • ...and 9 more figures

Theorems & Definitions (79)

  • Remark 1.1
  • Theorem 1.2
  • Remark 1.3
  • Example 1.4
  • Corollary 1.5
  • Theorem 1.6: Main result
  • Remark 1.7
  • Theorem 1.9
  • Remark 1.10
  • Definition 1.11
  • ...and 69 more