Abstract
We prove the existence of
minimal
and
rigid
singular holomorphic foliations by curves on the projective space
\mathbb{CP}^n
for every dimension
n≥ 2
and every degree
d ≥ 2
. Precisely, we construct a foliation
\mathcal 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
\mathbb{CP}^n
. Moreover,
\mathcal F
satisfies many additional properties expected from chaotic dynamics and is rigid in the following sense: if
\mathcal 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
\mathcal F
.
This is done by considering pseudo-groups generated on the unit ball
\mathbb{B}^n ⊂ \mathbb{C}^n
by small perturbations of elements in
\mathrm{Diff}(\mathbb{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
\mathbb{CP}^n
.