Abstract
Given \(r\geqslant 1\), the discrete \(C^r\)-centralizer of a vector field is formed by the set of its symmetries, that is, the set of \(C^r\)-diffeomorphisms commuting with the flow generated by it. Here we prove that if M is a compact surface and \(2\leqslant r \leqslant \infty \) then there exists a \(C^r\)-open and dense subset of vector fields \(\mathcal O \subset \mathfrak {X}^r(M)\) whose \(C^r\)-discrete centralizer is abelian. Moreover, we show that every symmetry of the flow \((X_t)_{t\in \mathbb R}\) generated by \(X\in \mathcal O\) is isomorphic to the direct product of the time-t maps of the flow and an abelian subgroup formed by involutions. In particular, we deduce that there exist \(C^1\)-open sets of \(C^r\)-Morse-Smale vector fields on surfaces whose \(C^r\)-discrete centralizer is trivial.
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Acknowledgements
The authors are deeply grateful to the anonymous referee for very useful comments. JR and PV were partially supported by CMUP, which is financed by national funds through FCT - Fundação para a Ciência e a Tecnologia, I.P., under the project with reference UIDB/00144/2020. PV also acknowledge financial support from the project PTDC/MAT-PUR/4048/2021 and benefited from the grant CEECIND/03721/2017 of the Stimulus of Scientific Employment, Individual Support 2017 Call, awarded by FCT, and from the project PTDC/MAT-PUR/29126/2017.
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Bonomo, W., Rocha, J. & Varandas, P. Symmetries of \(C^r\)-vector Fields on Surfaces. Bull Braz Math Soc, New Series 54, 45 (2023). https://doi.org/10.1007/s00574-023-00361-9
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DOI: https://doi.org/10.1007/s00574-023-00361-9