Symmetries of vector fields: The diffeomorphism centralizer

Davi Obata

doi:10.3934/dcds.2021063

Article Contents

2021, Volume 41, Issue 10: 4943-4957. Doi: 10.3934/dcds.2021063

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Symmetries of vector fields: The diffeomorphism centralizer

Davi Obata

Department of Mathematics, The University of Chicago, Chicago, IL, 60637, USA

Received: August 31, 2020

Revised: January 31, 2021

Early access: March 2021

Published: October 2021

D.O. was supported by the ERC project 692925 NUHGD

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In this paper we study the diffeomorphism centralizer of a vector field: given a vector field it is the set of diffeomorphisms that commutes with the flow. Our main theorem states that for a $ C^1 $-generic diffeomorphism having at most finitely many sinks or sources, the diffeomorphism centralizer is quasi-trivial. In certain cases, we can promote the quasi-triviality to triviality. We also obtain a criterion for a diffeomorphism in the centralizer to be a reparametrization of the flow.

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Mathematics Subject Classification: 37C10, 37C20, 37C80, 37D30.

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