Orbital Linearization of Smooth Completely Integrable Vector Fields

TL;DR

This paper proves a smooth local orbital linearization theorem for integrable vector fields near nondegenerate singular points, using formal linearization, blowing-up, and Sternberg-Chen theorems.

Contribution

It establishes the orbital linearization for smooth integrable vector fields near singular points, extending previous formal results to the smooth category.

Findings

Proves smooth orbital linearization theorem for integrable vector fields.

Utilizes blowing-up and Sternberg-Chen isomorphism techniques.

Bridges formal and smooth linearization theories.

Abstract

The main purpose of this paper is to prove the smooth local orbital linearization theorem for smooth vector fields which admit a complete set of first integrals near a nondegenerate singular point. The main tools used in the proof of this theorem are the formal orbital linearization theorem for formal integrable vector fields, the blowing-up method, and the Sternberg-Chen isomorphism theorem for formally-equivalent smooth hyperbolic vector fields.