On the conservative pasting lemma

TL;DR

This paper develops perturbation tools for volume-preserving vector fields and diffeomorphisms, enabling pasting, smoothing, and linearization while maintaining regularity and volume constraints.

Contribution

It introduces new perturbation techniques in the volume-preserving setting, including a conservative pasting lemma and a linearized Franks lemma applicable across various regularity classes.

Findings

Pasting and local linearization hold in all regularity classes from C^1 to C^∞.

A conservative linearized Franks lemma is proved for C^{r,α} and C^∞ diffeomorphisms.

The resulting diffeomorphisms preserve the original regularity and volume constraints.

Abstract

Several perturbation tools are established in the volume preserving setting allowing for the pasting, extension, localized smoothing and local linearization of vector fields. The pasting and local linearization hold in all classes of regularity ranging from $C^{1}$ to $C^{\infty}$ (H\"older included). For diffeomorphisms, a conservative linearized version of Franks lemma is proved in the $C^{r,\alpha}$ ($r\in\mathbb{Z}^+$, $0<\alpha <1$) and $C^{\infty}$ settings, the resulting diffeomorphism having the same regularity as the original one.