[Paper Review] The 2/3 - convergence rate for the Poisson bracket

Lev Buhovsky|arXiv (Cornell University)|Feb 26, 2008

Holomorphic and Operator Theory4 references4 citations

TL;DR

This paper introduces a novel method to establish $C^0$-rigidity of the Poisson bracket on symplectic manifolds, proving a precise $2/3$-H"older modulus of continuity for the supremum norm of the Poisson bracket under uniform perturbations. The key result is a sharp upper bound on the rate of convergence of the Poisson bracket's supremum, derived via symplectic displacement energy estimates and differential operator analysis at maximum points of the bracket.

ABSTRACT

In this paper we introduce a new method for approaching the C^0 - rigidity results for the Poisson bracket. Using this method, we provide a different proof for the lower semi-continuity under C^0 perturbations, for the uniform norm of the Poisson bracket. We find the precise rate for the modulus of the semi-continuity. This extends the previous results of Cardin-Viterbo, Zapolsky, Entov and Polterovich. Using our method, we prove a C^0 - rigidity result in the spirit of the work of Humiliere. We also discuss a general question of the C^0 - rigidity for multilinear differential operators.

Motivation & Objective

• To establish a sharp quantitative $C^0$-rigidity result for the Poisson bracket under uniform perturbations.
• To provide a new proof of the lower semi-continuity of the uniform norm of the Poisson bracket, extending prior results by Cardin–Viterbo and Entov–Polterovich.
• To analyze the precise rate of convergence of the Poisson bracket's supremum under $C^0$-perturbations, identifying the $2/3$-power law as optimal.
• To investigate the non-local nature of $C^0$-rigidity by constructing counterexamples where perturbations affect the Poisson bracket globally despite local convergence.

Proposed method

• The method relies on symplectic displacement energy estimates, particularly using the Gromov width to bound the energy of embedded sets in Darboux charts.
• It introduces the function $ \Upsilon_{f,g}^{+}(\varepsilon) $, measuring the deficit in the supremum of $ \{F,G\} $ under $ \varepsilon $-uniform perturbations of $ f,g $, and derives its asymptotic behavior as $ \varepsilon \to 0 $.
• The analysis focuses on points where $ \{f,g\} $ attains its maximum, assuming non-criticality of $ f $ and $ g $ at such points to ensure non-degeneracy.
• A key technical tool is the use of higher-order differential operators $ \mathcal{D}^l $ applied to $ \{f,g\} $, leading to bounds involving $ \mathcal{D}^l(\{f,g\})(x) $ at the maximum point.
• The proof uses the completeness of Hamiltonian flows (i.e., $ G \in \mathcal{H}^b(M,\omega) $) to ensure global control over the dynamics during perturbations.
• The method generalizes to higher-order rigidity by analyzing the $ \varepsilon^{2l/(2l+1)} $-rate of convergence, with explicit constants derived from geometric and analytic estimates.

Experimental results

Research questions

• RQ1 What is the optimal rate of convergence for the supremum norm of the Poisson bracket under $ C^0 $-perturbations of the generating functions?
• RQ2 Can the $ C^0 $-rigidity of the Poisson bracket be quantified with a sharp modulus of continuity, and if so, what is its exact form?
• RQ3 To what extent is the $ C^0 $-rigidity of the Poisson bracket a local phenomenon, or does it depend on global dynamics?
• RQ4 How do displacement energy estimates and higher-order derivatives of the Poisson bracket influence the rigidity rate?
• RQ5 Can the rigidity result be extended to multilinear differential operators beyond the Poisson bracket?

Key findings

• The paper establishes a sharp upper bound on the convergence rate of the Poisson bracket's supremum: $ \limsup_{\varepsilon \to 0} \frac{\Upsilon_{f,g}^{+}(\varepsilon)}{\varepsilon^{2/3}} \leq 6 \left( -\{\{\{f,g\},f\},f\}(x) - \{\{\{f,g\},g\},g\}(x) \right)^{1/3} $, where $ x $ is a maximum point of $ \{f,g\} $.
• The $ 2/3 $-power rate is optimal and cannot be improved under the given assumptions, as shown by the asymptotic scaling derived from displacement energy and differential operator estimates.
• The method provides a new, shorter proof of the $ C^0 $-lower semi-continuity of $ \|\{f,g\}\| $, recovering and refining results from Entov–Polterovich and Zapolsky.
• The paper demonstrates that $ C^0 $-rigidity of the Poisson bracket is inherently non-local: perturbations can destroy the bracket's value at a point even if functions converge uniformly, due to fast Hamiltonian flows.
• For higher-order rigidity, the convergence rate is $ \varepsilon^{2l/(2l+1)} $, with explicit constants involving $ \mathcal{D}^l(\{f,g\})(x) $, showing a systematic hierarchy of rigidity rates.
• A counterexample is constructed where $ f_n \to f $, $ g_n \to g $ uniformly, but $ \{f_n, g_n\} \equiv 0 $ while $ \{f,g\} \equiv 1 $, illustrating the necessity of the flow completeness condition $ G \in \mathcal{H}^b(M,\omega) $.