On the failure of linearization for germs of $C^1$ hyperbolic vector fields in dimension one

TL;DR

This paper demonstrates that classical linearization theorems for hyperbolic 1D vector fields do not hold in low regularity settings, providing explicit examples of non-conjugate flows with identical multipliers.

Contribution

It shows the failure of Sternberg's linearization theorem in low regularity by constructing explicit non-conjugate hyperbolic flows with the same multipliers.

Findings

Classical linearization fails in low regularity settings.

Explicit uncountable families of non-conjugate flows are constructed.

Conjugacy classes are distinguished in bi-Lipschitz, C^1, and C^{1+ac} categories.

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.