Minimal, rigid foliations by curves on
Frank Loray
Université Lille I, Villeneuve D'ascq, FranceJulio C. Rebelo
Université Toulouse 3, France
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Abstract
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