The real-time trajectories of a homogeneous holomorphic vector field in ${\mathbb C}^2$ project to geodesics of some affine structure on ${\mathbb CP}^1$. We investigate the asymptotic behavior of such geodesics.
It is known that the $\omega$-limit set of such a geodesic may be a periodic geodesic, may be a union of connections between singularities or may be dense in the Riemann sphere. We prove the existence of geodesics whose $\omega$-limit set is locally the product of a Cantor set with an interval (such a possibility was already known for geodesics of affine structures on higher genus Riemann surfaces).
