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

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

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 $\mathcal{F}$ is conjugate to some $\mathcal{F}'\in\mathcal{U}$ 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 $\text{Diff}(\mathbb{C}^n,0)$ on a fixed ball.