Trajectories of vector fields asymptotic to formal invariant curves

Abstract

We prove that a formal curve $\Gamma$ that is invariant by a $C^\infty$ vector field $\xi$ of $\mathbb{R}^m$ has a geometrical realization, as soon as the Taylor expansion of $\xi$ is not identically zero along $\Gamma$. This means that there is a trajectory $\gamma$ of $\xi$ which is asymptotic to $\Gamma$. 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 $\Gamma$ which is composed entirely of trajectories asymptotic to $\Gamma$, and contains the germ of any such trajectory. If $\xi$ is analytic, we prove that there exists a trajectory asymptotic to $\Gamma$ which is, moreover, non-oscillating with respect to subanalytic sets.