Institute for Mathematical Sciences

Preprint ims00-05

F. Loray and J. Rebelo
Stably chaotic rational vector fields on $\Bbb C\Bbb P^n$.

Abstract: We construct an open set $\Cal U$ of rational foliations of arbitrarily fixed degree $d \ge 2$ by curves in $\Bbb C\Bbb P^n$ such that any foliation $\Cal F\in\Cal U$ has a finite number of singularities and satisfies the following chaotic properties.

Minimality: any leaf (curve) is dense in $\Bbb C\Bbb P^n$.

Ergodicity: any Lebesgue measurable subset of leaves has zero or total Lebesgue measure.

Entropy: the topological entropy is strictly positive even far from singularities.

Rigidity: if $\Cal F$ is conjugate to some $\Cal F'\in\Cal U$ by a homeomorphism close to the identity, then they are also conjugate by a projective transformation.

The main analytic tool employed in the construction of these foliations is the existence of several pseudo-flows in the closure of pseudo-groups generated by perturbations of elements in $\text{Diff}(\Bbb C^n,0)$ on a fixed ball.

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