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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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References:
 [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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