On the non-persistence of Hamiltonian identity cycles

TL;DR

This paper investigates the conditions under which Hamiltonian identity cycles do not persist under perturbations, demonstrating that for generic cases and specific cycles, the leading term of the holonomy map is non-zero, indicating non-persistence.

Contribution

It proves the non-persistence of Hamiltonian identity cycles for generic perturbations and certain cycles using Chen's iterated integrals theory.

Findings

Non-vanishing of the leading holonomy term implies cycle non-persistence

Generic perturbations lead to non-persistence of cycles

Cycles as commutators in degree two Hamiltonian foliations also non-persistent

Abstract

We study the leading term of the holonomy map of a perturbed plane polynomial Hamiltonian foliation. The non-vanishing of this term implies the non-persistence of the corresponding Hamiltonian identity cycle. We prove that this does happen for generic perturbations and cycles, as well for cycles which are commutators in Hamiltonian foliations of degree two. Our approach relies on the Chen's theory of iterated path integrals which we briefly resume.