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Minimal, rigid foliations by curves on ℂℙ n

Minimal, rigid foliations by curves on ℂℙ n 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 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of the European Mathematical Society Springer Journals

Minimal, rigid foliations by curves on ℂℙ n

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Publisher
Springer Journals
Copyright
Copyright © 2003 by Springer-Verlag Berlin Heidelberg & EMS
Subject
Mathematics; Mathematics, general
ISSN
1435-9855
DOI
10.1007/s10097-002-0049-6
Publisher site
See Article on Publisher Site

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 .

Journal

Journal of the European Mathematical Society – Springer Journals

Published: Jun 1, 2003

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