[Paper Review] C^0-rigidity of the double Poisson bracket

Michael Entov, Leonid Polterovich|ArXiv.org|Jul 27, 2008

Geometric and Algebraic Topology12 references3 citations

TL;DR

This paper establishes a $C^0$-rigidity dichotomy for functionals built from the double Poisson bracket on symplectic manifolds. Using Hofer geometry and Landau-Hadamard-Kolmogorov inequalities, it proves that such functionals are either weakly robust but not lower semicontinuous (when only one pair of functions is involved), or fully lower semicontinuous (when both pairs contribute positively), with sharp convergence rates in the latter case. The result reveals a hierarchy of robustness in symplectic function theory under uniform perturbations.

ABSTRACT

The paper is devoted to function theory on symplectic manifolds. We study a natural class of functionals involving the double Poisson brackets from the viewpoint of their robustness properties with respect to small perturbations in the uniform norm. We observe an hierarchy of such robustness properties. The methods involve Hofer's geometry on the symplectic side and Landau-Hadamard-Kolmogorov inequalities on the function-theoretic side.

Motivation & Objective

• To investigate the robustness of functionals built from the double Poisson bracket under $C^0$-small perturbations on symplectic manifolds.
• To classify these functionals based on their lower semicontinuity and weak robustness properties.
• To establish sharp convergence rates for lower semicontinuous functionals using geometric and analytic tools.
• To extend known results on Poisson bracket rigidity to higher-order iterated brackets and identify open problems in the general case.

Proposed method

• Uses Hofer’s geometry on the symplectic side to bound the Hofer norm of Hamiltonian flows generated by Lie polynomials.
• Applies Landau-Hadamard-Kolmogorov inequalities to control higher-order derivatives in terms of lower-order norms, especially for iterated Poisson brackets.
• Constructs explicit counterexamples via oscillatory perturbations $F_N = F + \frac{1}{N}a\sin(NF)$ to demonstrate failure of lower semicontinuity in the weakly robust case.
• Re-proves Buhovsky’s $2/3$-law for the functional $\max\{F,G\}$ using geometric methods, extending the framework to double brackets.
• Employs a flow-based approach $u_t = \prod \phi^{t}_{a_j F + b_j G}$ to derive tight bounds on the Hofer norm, linking it to maxima and minima of Lie monomials.
• Applies Kolmogorov’s inequality to $\text{ad}_F^N G$ to prove weak robustness for iterated adjoint functionals.

Experimental results

Research questions

• RQ1 Under what conditions is a functional built from the double Poisson bracket weakly robust or lower semicontinuous under $C^0$-perturbations?
• RQ2 Can sharp convergence rates be established for lower semicontinuous functionals involving the double bracket?
• RQ3 Is the functional $\text{osc}(\text{ad}_F)^N G$ lower semicontinuous for $N \geq 3$?
• RQ4 Can the method used for $N=2$ be extended to prove failure of lower semicontinuity for higher-order iterated brackets?
• RQ5 Do new geometric or analytic tools exist that can bound the Hofer norm of flows generated by general Lie polynomials in two variables?

Key findings

• The functional $\Phi^v(F,G) = v_1 \max\{\{F,G\},F\} - v_2 \min\{\{F,G\},F\} + v_3 \max\{\{F,G\},G\} - v_4 \min\{\{F,G\},G\}$ is weakly robust but not lower semicontinuous if only $v_1,v_2$ or only $v_3,v_4$ are positive.
• If both $v_1,v_2 > 0$ and $v_3,v_4 > 0$, then $\Phi^v$ is lower semicontinuous, indicating stronger stability under $C^0$-perturbations.
• A sharp $2/3$-law is recovered: $\Phi(F,G) - \overline{\Phi}_\epsilon(F,G) \leq C \cdot \Psi(F,G)^{1/3} \epsilon^{2/3}$, where $\Psi(F,G) = \|\{\{\{F,G\},F\},F\} + \{\{\{F,G\},G\},G\}\|$.
• The functional $\max\{\{F,G\},F\}$ is weakly robust due to the Landau-Hadamard inequality: $\max\{\{F,G\},F\} \geq \frac{\|\{F,G\}\|^2}{2 \cdot \text{osc}\, G}$.
• For any $N \in \mathbb{N}$, the functional $\Phi(F,G) = \text{osc}(\text{ad}_F)^N G$ is weakly robust, as shown via Kolmogorov’s inequality: $\text{osc}\, v^{(N)} \geq C_N \frac{\|v'\|^N}{\|v\|^{N-1}}$.
• The functional $\Phi_{k,m}(F,G) = \text{osc}(\text{ad}_H)^m G$ with $H = (\text{ad}_G)^k F$ satisfies $\Phi_{k,m}(F,G) \geq C_{k,m} \frac{\|\{F,G\}\|^{(k+1)m}}{\|F\|^{km} \|G\|^{m-1}}$, proving its weak robustness.