Simultaneous local normal forms of dynamical systems with singular underlying geometric structures (Q6608796)

A classical result in Hamiltonian theory says that a Hamiltonian vector field \(X\) which vanishes at a point has a normalization at that point. This means that there is a coordinate system \((x_1, \dots, x_{2n})\) which contains no nonresonant terms and the symplectic form has the canonical form \(\omega = \sum _{i=1}^{n} dx_i \wedge dx_{n+i}\). A similar result holds for volume-preserving vector fields.\N\NIn this paper, the authors consider the problem of normalizing Hamiltonian and volume-preserving vector fields as simultaneous normalization of a pair \((X, \mathcal{G})\), where \(\mathcal{G}\) is an underlying geometric structure preserved by \(X\). Note that \(\mathcal{G}\) can be a volume form, a symplectic form, a contact form, a Poisson tensor, or their singular versions. Simultaneous normalization is difficult to achieve, particularly when \(X\) and \(\mathcal{G}\) are singular, even when the normal forms of \(X\) and \(\mathcal{G}\) can be established independently.\N\NThe goal of this paper is to describe how to achieve simultaneous normalization when the geometric structure \(\mathcal{G}\) is singular at a singular point \(O\) of \(X\) (so that \(X(O) = 0\).) The authors impose some nondegeneracy conditions so that the singular geometrical structure itself has something like a nice local canonical expression. They extend the method first described by the third author in [Arch. Ration. Mech. Anal 229, 789--833 (2018; Zbl 1398.37053)] by showing that it also applies to singular structures.\N\NThe theory that the authors develop is based on a new approach, called by them ``toric conservation principle'', as well as Moser's equivariant path method and the classical stepwise normalization technique.