On the Failure of Linearization for Germs of c1 Hyperbolic Vector Fields in Dimension One

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, C1 and C1+ac settings. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2023.