# Minimal, rigid foliations by curves on $\u2102{\mathbb{P}}^{n}$

Journal of the European Mathematical Society (2003)

- Volume: 005, Issue: 2, page 147-201
- ISSN: 1435-9855

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topLoray, 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 -

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