# Minimal, rigid foliations by curves on ℂℙ<sup><var>n</var></sup>

### Frank Loray

Université Lille I, Villeneuve D'ascq, France### Julio C. Rebelo

Université Toulouse 3, France

## 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.

## Cite this article

Frank Loray, Julio C. Rebelo, Minimal, rigid foliations by curves on ℂℙ<sup><var>n</var></sup>. J. Eur. Math. Soc. 5 (2003), no. 2, pp. 147–201

DOI 10.1007/S10097-002-0049-6