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

Abstract

Let ξ be an analytic vector field at (R3, 0) and I be an analytically non-oscillatory integral pencil of ξ; i.e., I is a maximal family of analytically non-oscillatory trajectories of ξ at 0 all sharing the same iterated tangents. We prove that if I is interlaced, then for any trajectory Γ ∈ I, the expansion Ran,Γ of the structure Ran by Γ is model-complete, o-minimal and polynomially bounded.