Minimal, rigid foliations by curves on ${\mathbb{CP}}^n$

Abstract

We prove the existence of minimal and rigid singular holomorphic foliations by curves on the projective space ${\mathbb{CP}}^n$ for every dimension $n\ge 2$ and every degree $d\ge 2$. Precisely, we construct a foliation $\mathcal{F}$ which is induced by a homogeneous vector field of degree $d$, has a finite singular set and all the regular leaves are dense in the whole of ${\mathbb{CP}}^n$. Moreover, $\mathcal{F}$ satisfies many additional properties expected from chaotic dynamics and is rigid in the following sense: if $\mathcal{F}$ is conjugate to another holomorphic foliation by a homeomorphism sufficiently close to the identity, then these foliations are also conjugate by a projective transformation. Finally, all these properties are persistent for small perturbations of $\mathcal{F}$.

This is done by considering pseudo-groups generated on the unit ball ${\mathbb{B}}^n\subset {\mathbb{C}}^n$ by small perturbations of elements in $\mathrm{Diff}({\mathbb{C}}^n,0)$. Under open conditions on the generators, we prove the existence of many pseudo-flows in their closure (for the $C^0$-topology) acting transitively on the ball. Dynamical features as minimality, ergodicity, positive entropy and rigidity may easily be derived from this approach. Finally, some of these pseudo-groups are realized in the transverse dynamics of polynomial vector fields in ${\mathbb{CP}}^n$.