Structural stability of the Jouanolou foliations in every degree

Abstract

We show that the degree $d$ Jouanolou foliation ${\mathcal{J}}_d$ on ${\mathbb{P}}^2$ is structurally stable for every $d\ge 2$. We prove the existence of a neighbourhood ${\mathcal{U}}_d$ of ${\mathcal{J}}_d$, such that every foliation $\mathcal{F}\in {\mathcal{U}}_d$ satisfies: the Fatou set is a connected disc bundle over a compact Riemann surface of genus $d(d+1)/2$, no leaf is dense in ${\mathbb{P}}^2$, and all but countably many regular leaves are biholomorphic to the unit disc. Finally, we prove transverse perfectness of the Julia set on the regular locus for $d\ge 2$.

arXiv comment27 pages. Comments are welcome