Solitary trajectories of analytic vector fields

Abstract

A trajectory $$\gamma $$ of an analytic vector fields $$\xi $$ is said to be solitary, if $$\gamma $$ is asymptotic to a formal curve and can be isolated from any other trajectory of $$\xi $$ by a finite sequence of point centered blow ups. We prove that the collection of all solitary trajectories generates a polynomially bounded o-minimal structure.