Dynamical aspects of foliations with ample normal bundle

We prove the following result that was conjectured by Brunella: Let $X$ be a compact complex manifold of dimension $\ge 3$. Let $\mathcal{F}$ be a codimension one holomorphic foliation on $X$ with ample normal bundle. Then every leaf of $\mathcal{F}$ accumulates to the singular set of $\mathcal{F}$.