New results on the linearization of Nambu structures

TL;DR

This paper extends the classification and linearization results of Nambu structures, proving smooth linearizability for previously unresolved hyperbolic cases and strengthening analytic linearization theorems.

Contribution

It demonstrates smooth linearization for Type 1 hyperbolic Nambu structures and improves the analytic linearization theorem from prior work.

Findings

Hyperbolic Nambu structures are smoothly linearizable.

Enhanced analytic linearization theorem established.

Completes classification of linear Nambu structures.

Abstract

In a paper with Jean-Paul Dufour in 1999 \cite{DufourZung-Nambu1999}, we gave a classification of linear Nambu structures, and obtained linearization results for Nambu structures with a nondegenerate linear part. There was a case left open in \cite{DufourZung-Nambu1999}, namely the case of smooth linearization of Nambu structures with a Type 1 hyperbolic linear part which satisfies a natural signature condition. In this paper, we will show that such hyperbolic Nambu structures are also smoothly linearizable. We will also give a strong version of the analytic linearization theorem in the analytic case, improving a result obtained in \cite{DufourZung-Nambu1999}.