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 ℂℙn for every dimension n ≥ 2 and every degree d ≥ 2. Precisely, we construct a foliation 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 ℂℙn. Moreover, ℱ satisfies many additional properties expected from chaotic dynamics and is rigid in the following sense: if 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 ℱ. This is done by considering pseudo-groups generated on the unit ball Bn ⊂ ℂn by small perturbations of elements in Diff(ℂn,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 ℂℙn.