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On the conservative pasting lemma

Part of: Smooth dynamical systems: general theory

Published online by Cambridge University Press:  17 October 2018

PEDRO TEIXEIRA
PEDRO TEIXEIRA*
Affiliation:
Centro de Matemática da Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal email [email protected]
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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.

Keywords

divergence-free vector fieldlocalized smoothinglocal linearizationvolume-preserving diffeomorphismsFranks’ lemma

MSC classification

Primary: 37C10: Vector fields, flows, ordinary differential equations
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 5 , May 2020 , pp. 1402 - 1440
Copyright
© Cambridge University Press, 2018

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