Limit sets within curves where trajectories converge to

For continuously differentiable vector fields, we characterize the ω limit set of a trajectory converging to a compact curve Γ⊂Rn. In particular, the limit set is either a fixed point or a continuum of fixed points if Γ is a simple open curve; otherwise, the limit set can in addition be either a closed orbit or a number of fixed points with compatibly oriented orbits connecting them. An implication of the result is a tightened-up version of the Poincaré-Bendixson theorem.