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Minimal, rigid foliations by curves on \(\mathbb C\mathbb P^n\). (English) Zbl 1021.37030
In this very nice paper the authors construct examples of minimal and rigid singular degree \(d\) holomorphic foliations. \(F\), on the projective space \(\mathbb{C}\mathbb{P}^n\) for every \(n\geq 2\) and every degree \(d\geq 2\). Their foliation \(F\) has a finite singular set, all regular leaves are dense in the whole \(\mathbb{C}\mathbb{P}^n\). Furthermore, \(F\) satisfies many additional properties expected from chaotic dynamics (ergodicity: every measurable set of leaves has zero or total Lebesgue measure) and a very strong rigidity property: 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 \(F\) and in particular they find an open subset of the degree \(d\) foliations having all these properties.
To construct \(F\) they consider pseudo-groups generated on the unit ball of \(\mathbb{C}^n\) by the germs at \(0\in \mathbb{C}^n\) of diffeomorphisms. On this topic they prove important results on the existence of many pseudo-flows.

MSC:
37F75 Dynamical aspects of holomorphic foliations and vector fields
32M25 Complex vector fields, holomorphic foliations, \(\mathbb{C}\)-actions
58H05 Pseudogroups and differentiable groupoids
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[1] Arnold, V.I., Il’yashenko, Yu.S.: Ordinary differential equations. In: Encyclo- pedia of Math. Sciences Vol. 1, Dynamical Systems I, Anosov, D.V., Arnold, V.I. (eds.), pp. 1-148. Springer 1988 · Zbl 0659.58012
[2] Belliart, M.: Sur certains pseudogroupes de biholomorphismes locaux de (Cn, 0). Bull. Soc. Math. Fr. 129 , 259-284 (2001) · Zbl 1006.58015 · smf.emath.fr
[3] Belliart, M.: On the dynamics of certain actions of free groups on closed real analytic manifolds. Comment. Math. Helv. 77 , 524-548 (2002) · Zbl 1140.37312 · doi:10.1007/s00014-002-8350-2
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