Minimal, rigid foliations by curves on ??
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Authors: | Frank Loray Julio C Rebelo |
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Institution: | UMR du CNRS?8524, U.F.R. de Mathématiques, Université Lille?I, 59655?Villeneuve d’Ascq Cedex, France; e-mail:?loray@agat.univ-lille1.fr, FR Permanent address: Pontificia Universidade Catolica do Rio de Janeiro PUC-Rio, Rua Marquês de S?o Vicente 225 – Gávea, Rio de Janeiro RJ Brasil CEP?22453-900; e-mail:?jrebelo@mat.puc-rio.br, Current address: IMS – Math. Tower, State University of New York at Stony Brook, Stony Brook N.Y. 11794 – 3660 USA; e-mail:?jrebelo@math.sunysb.edu, BR
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Abstract: | We prove the existence of minimal and rigid singular holomorphic foliations by curves on the projective space ℂℙ
n
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 ℂℙ
n
. 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 𝔹
n
⊆ℂ
n
by small perturbations of elements in Diff(ℂ
n
,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 ℂℙ
n
.
Received March 7, 2002 / final version received November 26, 2002?Published online February 7, 2003
Most of this work has been carried out during a visit of the first author to IMPA/RJ and a visit of the second author to the
University of Lille 1. We would like to thank these institutes for hospitality and express our gratitude to CNPq-Brazil and
CNRS-France for the financial support which made these visits possible. We are also indebted to Paulo Sad, Marcel Nicolau
and the referee whose comments helped us to improve on the preliminary version. Finally, the second author has partially conducted
this research for the Clay Mathematics Institute. |
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Keywords: | |
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