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Minimal, rigid foliations by curves on ℂℙn

Authors
  • Loray, Frank1
  • Rebelo, Julio C.2
  • 1 UMR du CNRS 8524, U.F.R. de Mathématiques, Université Lille I, 59655 Villeneuve d’Ascq Cedex, France; e-mail: [email protected], FR
  • 2 Permanent address: Pontificia Universidade Catolica do Rio de Janeiro PUC-Rio, Rua Marquês de São Vicente 225 – Gávea, Rio de Janeiro RJ Brasil CEP 22453-900; e-mail: [email protected], Current address: IMS – Math. Tower, State University of New York at Stony Brook, Stony Brook N.Y. 11794 – 3660 USA; e-mail: [email protected], BR
Type
Published Article
Journal
Journal of the European Mathematical Society
Publisher
Springer-Verlag
Publication Date
Jun 01, 2003
Volume
5
Issue
2
Pages
147–201
Identifiers
DOI: 10.1007/s10097-002-0049-6
Source
Springer Nature
Keywords
License
Yellow

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 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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