Minimal, rigid foliations by curves on

  • Frank Loray

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

    Université Toulouse 3, France


We prove the existence of minimal and rigid singular holomorphic foliations by curves on the projective space for every dimension and every degree . Precisely, we construct a foliation which is induced by a homogeneous vector field of degree , has a finite singular set and all the regular leaves are dense in the whole of . 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 by small perturbations of elements in . Under open conditions on the generators, we prove the existence of many pseudo-flows in their closure (for the -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 .

Cite this article

Frank Loray, Julio C. Rebelo, Minimal, rigid foliations by curves on . J. Eur. Math. Soc. 5 (2003), no. 2, pp. 147–201

DOI 10.1007/S10097-002-0049-6