Abstract
We investigate conjugacy classes of germs of hyperbolic 1-dimensional vector fields at the origin in low regularity. We show that the classical linearization theorem of Sternberg strongly fails in this setting by providing explicit uncountable families of mutually non-conjugate flows with the same multipliers, where conjugacy is considered in the bi-Lipschitz, \(C^1\) and \(C^{1+ac}\) settings.
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As a straightforward computation shows, Sternberg’s germ of diffeomorphism (1) has absolutely continuous derivative as well; by [4], it embeds into a \(C^1\) flow (we do not know whether it embeds into a \(C^{1+ac}\) flow, but it follows also from [4] that it is at least \(C^1\) conjugate to a diffeomorphism that embeds in such a flow). However, Sternberg proved that it is not bi-Lipschitz conjugate to its linear part.
This example for \(\alpha = 1\) was already considered in [12, Exercise 4.1.12] However, our approach here is much more concrete (and is not based only on a “miraculous” direct integration).
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Acknowledgements
We would like to thank Étienne Ghys for having asked the question of existence of non-linearizable hyperbolic vector fields of class \(C^1\) in relation to his work with Grant Cairns on linearization of local \(\textrm{SL}(n)\)-actions [5]. We would also like to thank Jan Kiwi for his interest on the subject, as well as Frédéric Le Roux for several clever remarks. Some material of this work arose following a question of Michele Triestino about normalizers of flows near a singularity and we thank him for stimulating discussions. H. Eynard-Bontemps was detached at the CMM of University of Chile during the elaboration of this work and was partially funded by CNRS and by the IRGA project ADMIN of Grenoble INP - Université Grenoble Alpes. A. Navas was supported by the Fondecyt research project 1220032.
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Hélène Eynard-Bontemps was partially funded by CNRS (Centre National de Recherche Scientifique, France), IRGA project ADMIN (Grenoble INP - Université Grenoble Alpes), Centro de Modelamiento Matemático (CMM, FB210005, BASAL funds for centers of excellence from ANID-Chile). Andrés Navas was supported by Fondecyt research project 1220032.
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HE-B and AN have contributed equally to the work.
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Eynard-Bontemps, H., Navas, A. On the Failure of Linearization for Germs of \(C^1\) Hyperbolic Vector Fields in Dimension One. J Dyn Diff Equat (2023). https://doi.org/10.1007/s10884-023-10330-x
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DOI: https://doi.org/10.1007/s10884-023-10330-x