Geodesics in affine surfaces

par Xavier Buff (Institut de Mathématiques de Toulouse)

vendredi 20 oct. 2023, 10:30 → 11:30 Europe/Paris

207 (Bat 1R2)

The real-time trajectories of a homogeneous holomorphic vector field in ${\mathbb{C}}^2$ project to geodesics of some affine structure on ${\mathbb{C}P}^1$. 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).