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Trajectories of vector fields asymptotic to formal invariant curves
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems , First View
- Published online by Cambridge University Press:
- 01 April 2026, pp. 1-32
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- Article
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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 
$\gamma \subset \mathbb R^{m}$ of 
$\xi $ which is asymptotic to 
$\Gamma $. This result solves a natural question proposed by Bonckaert [Smooth invariant curves of singularities of vector fields in R3. Ann. Inst. Henri Poincaré 3(2) (1986), 111–183] 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 
$\gamma $ asymptotic to 
$\Gamma $ which is, moreover, non-oscillating with respect to subanalytic sets.
