Trajectories of vector fields asymptotic to formal invariant curves

Abstract

We prove that a formal curve $Γ$ that is invariant by a $C^\infty$ vector field $ξ$ of $\mathbb{R}^m$ has a geometrical realization, as soon as the Taylor expansion of $ξ$ is not identically zero along $Γ$. This means that there is a trajectory $γ$ of $ξ$ which is asymptotic to $Γ$. This result solves a natural question proposed by Bonckaert nearly forty years ago. We also construct an invariant $C^0$ manifold $S$ in some open horn around $Γ$ which is composed entirely of trajectories asymptotic to $Γ$, and contains the germ of any such trajectory. If $ξ$ is analytic, we prove that there exists a trajectory asymptotic to $Γ$ which is, moreover, non-oscillating with respect to subanalytic sets.