Trajectories in interlaced integral pencils of 3-dimensional analytic vector fields are o-minimal

Abstract:

Let $\xi$ be an analytic vector field at $(\mathbb {R}^3,0)$ and $\mathcal {I}$ be an analytically non-oscillatory integral pencil of $\xi$; i.e., $\mathcal {I}$ is a maximal family of analytically non-oscillatory trajectories of $\xi$ at 0 all sharing the same iterated tangents. We prove that if $\mathcal {I}$ is interlaced, then for any trajectory $\Gamma \in \mathcal {I}$, the expansion $\mathbb {R}_{\mathrm {an},\Gamma }$ of the structure $\mathbb {R}_{\mathrm {an}}$ by $\Gamma$ is model-complete, o-minimal and polynomially bounded.