On the failure of linearization for germs of C¹ hyperbolic vector fields in dimension one

2022/12/27 by Hélène Eynard-Bontemps, Eynard-Bontemps, Hélène, Andrés Navas +1
Mathematics · Physics and Astronomy · #37D99 #37E05 #37G05 #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.2212.13646

openalex publication_date 2022/12/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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, C¹ and C^1+ac settings.

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