On the conservative pasting lemma

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 the local linearization hold in all classes of regularity ranging from$C^{1}$to$C^{\infty }$(Hölder included). For diffeomorphisms, a conservative linearized version of Franks’ lemma is proved in the$C^{r,\unicode[STIX]{x1D6FC}}$($r\in \mathbb{Z}^{+}$,$0<\unicode[STIX]{x1D6FC}<1$) and$C^{\infty }$settings, the resulting diffeomorphism having the same regularity as the original one.