Rigidity in topology C^0 of the Poisson bracket for Tonelli Hamiltonians

TL;DR

This paper establishes a rigidity result for the Poisson bracket of Tonelli Hamiltonians, showing that under certain convergence conditions, the limit of the brackets equals the bracket of the limits, highlighting stability in the C^0 topology.

Contribution

It proves a new rigidity theorem for the Poisson bracket in the C^0 topology for Tonelli Hamiltonians, extending understanding of their stability under convergence.

Findings

Poisson brackets converge to the bracket of limits under C^0 convergence.

The limit of the Poisson brackets is exactly the Poisson bracket of the limit Hamiltonians.

The result applies to sequences of Tonelli Hamiltonians on cotangent bundles.

Abstract

We prove the following rigidity result for the Tonelli Hamiltonians. Let T * M be the cotangent bundle of a closed manifold M endowed with its usual symplectic form. Let (F\_n) be a sequence of Tonelli Hamiltonians that C^0 converges on the compact subsets to a Tonelli Hamiltonian F. Let (G\_n) be a sequence of Hamiltonians that that C^0 converges on the compact subsets to a Hamiltonian G. We assume that the sequence of the Poisson brackets ({F\_n , G\_n }) C^0-converges on the compact subsets to a C^1 function H. Then H = {F, G}.