Minimal, rigid foliations by curves on $ℂ{\mathbb{P}}^n$

Loray, Frank, and Rebelo, Julio C.. "Minimal, rigid foliations by curves on $\mathbb {C}\mathbb {P}^n$." Journal of the European Mathematical Society 005.2 (2003): 147-201. <http://eudml.org/doc/277706>.

@article{Loray2003,
abstract = {We prove the existence of minimal and rigid singular holomorphic foliations by curves on the projective space $\mathbb \{C\}\mathbb \{P\}^n$ for every dimension $n\ge 2$ and every degree $d\ge 2$. 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 $\mathbb \{C\}\mathbb \{P\}^n$. 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 $\mathbb \{B\}^n\subset \mathbb \{C\}^n$ by small perturbations of elements in $\operatorname\{Diff\}(\mathbb \{C\}^n,0)$. 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 $\mathbb \{C\}\mathbb \{P\}^n$.},
author = {Loray, Frank, Rebelo, Julio C.},
journal = {Journal of the European Mathematical Society},
keywords = {meromorphic foliations; leaf of a foliation; dense leaf; polynomial vector fields; complex dynamics; ergodicity; holomorphic foliations; rigidity; meromorphic foliations; leaf of a foliation; dense leaf; polynomial vector fields; complex dynamics; ergodicity; holomorphic foliations; rigidity},
language = {eng},
number = {2},
pages = {147-201},
publisher = {European Mathematical Society Publishing House},
title = {Minimal, rigid foliations by curves on $\mathbb \{C\}\mathbb \{P\}^n$},
url = {http://eudml.org/doc/277706},
volume = {005},
year = {2003},
}

TY - JOUR
AU - Loray, Frank
AU - Rebelo, Julio C.
TI - Minimal, rigid foliations by curves on $\mathbb {C}\mathbb {P}^n$
JO - Journal of the European Mathematical Society
PY - 2003
PB - European Mathematical Society Publishing House
VL - 005
IS - 2
SP - 147
EP - 201
AB - We prove the existence of minimal and rigid singular holomorphic foliations by curves on the projective space $\mathbb {C}\mathbb {P}^n$ for every dimension $n\ge 2$ and every degree $d\ge 2$. 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 $\mathbb {C}\mathbb {P}^n$. 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 $\mathbb {B}^n\subset \mathbb {C}^n$ by small perturbations of elements in $\operatorname{Diff}(\mathbb {C}^n,0)$. 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 $\mathbb {C}\mathbb {P}^n$.
LA - eng
KW - meromorphic foliations; leaf of a foliation; dense leaf; polynomial vector fields; complex dynamics; ergodicity; holomorphic foliations; rigidity; meromorphic foliations; leaf of a foliation; dense leaf; polynomial vector fields; complex dynamics; ergodicity; holomorphic foliations; rigidity
UR - http://eudml.org/doc/277706
ER -