Journal of the European Mathematical Society

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Volume 5, Issue 2, 2003, pp. 147–201
DOI: 10.1007/s10097-002-0049-6

Minimal, rigid foliations by curves on ℂℙn

Frank Loray[1] and Julio C. Rebelo[2]

(1) UMR du CNRS 8524, Université Lille I, U.F.R. de Mathématiques, 59655, VILLENEUVE D'ASCQ CEDEX, FRANCE
(2) Institut de Mathématiques de Toulouse, Université Toulouse 3, 118 Route de Narbonne, 31062, TOULOUSE, FRANCE

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 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, satisfies many additional properties expected from chaotic dynamics and is rigid in the following sense: if 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 .

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 C0-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.

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Loray Frank, Rebelo Julio: Minimal, rigid foliations by curves on ℂℙn. J. Eur. Math. Soc. 5 (2003), 147-201. doi: 10.1007/s10097-002-0049-6