Another note on focus-focus singularities

TL;DR

This paper explores the relationship between monodromy formulas for focus-focus singularities in integrable Hamiltonian systems, extending previous results to degenerate cases and considering non-Hamiltonian scenarios and local normal forms.

Contribution

It establishes a connection between monodromy formulas and Duistermaat-Heckman formulas, extending prior work to degenerate and non-Hamiltonian cases.

Findings

Relation between monodromy and Duistermaat-Heckman formulas

Extension of results to degenerate focus-focus singularities

Analysis of non-Hamiltonian cases and local normal forms

Abstract

We show a natural relation between the monodromy formula for focus-focus singularities of integrable Hamiltonian systems and a formula of Duistermaat-Heckman, and extend the main results of our previous note on focus-focus singularities ($\bbS^1$-action, monodromy, and topological classification) to the degenerate case. We also consider the non-Hamiltonian case, local normal forms, etc.