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Dynamics of regularly ramified rational maps: Ⅰ. Julia sets of maps in one-parameter families
July
2018, 38(7): 3223-3237.
doi: 10.3934/dcds.2018140
Dicritical nilpotent holomorphic foliations
| 1. | Dpto. Ciencias - Sección Matemáticas, Pontificia Universidad Católica del Perú, Av. Universitaria 1801, San Miguel, Lima 32, Peru |
| 2. | Dpto. Álgebra, Análisis Matemático, Geometría y Topología, Facultad de Ciencias, Universidad de Valladolid, Paseo de Belén, 7; 47011 Valladolid, Spain |
| 3. | Dpto. Ciencias - Sección Matemáticas, Pontificia Universidad Católica del Perú, Av. Universitaria 1801, San Miguel, Lima 32, Peru |
We study in this paper several properties concerning singularities of foliations in $ {\left( {{\mathbb{C}}^{3}}\rm{,}\bf{0} \right)}$ that are pull-back of dicritical foliations in $ {\left( {{\mathbb{C}}^{2}}\rm{,}\bf{0} \right)}$. Particularly, we will investigate the existence of first integrals (holomorphic and meromorphic) and the dicriticalness of such a foliation. In the study of meromorphic first integrals we follow the same method used by R. Meziani and P. Sad in dimension two. While the foliations we study are pull-back of foliations in $ {\left( {{\mathbb{C}}^{2}}\rm{,}\bf{0} \right)}$, the adaptations are not straightforward.
Keywords:
Holomorphic foliations,
dicritical foliations.
Mathematics Subject Classification:
Primary: 37F75; Secondary: 32S65.
Citation:
Percy Fernández-Sánchez, Jorge Mozo-Fernández, Hernán Neciosup. Dicritical nilpotent holomorphic foliations. Discrete and Continuous Dynamical Systems,
2018, 38
(7)
: 3223-3237.
doi: 10.3934/dcds.2018140
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References:
| [1] |
C. Camacho,
Dicritical singularities of holomorphic vector fields, Contemporary Mathematics, 269 (2001), 39-45.
doi: 10.1090/conm/269/04328. |
| [2] |
C. Camacho and A. Lins Neto,
Teoria Geométrica das Folheações, IMPA, Projeto Euclides, 1979. |
| [3] |
F. Cano,
Reduction of the singularities of codimension one holomorphic foliations in dimension three, Annals of Math, 160 (2004), 907-1011.
doi: 10.4007/annals.2004.160.907. |
| [4] |
F. Cano, M. Ravara-Vago and M. Soares,
Local Brunella's alternative I. RICH foliations, Int. Math. Res. Not. IMRN, 9 (2015), 2525-2575.
doi: 10.1093/imrn/rnu011. |
| [5] |
F. Cano and D. Cerveau,
Desingularization of non-dicritical holomorphic foliations and existence of separatrices, Acta Math., 169 (1992), 1-103.
doi: 10.1007/BF02392757. |
| [6] |
D. Cerveau and J. -F. Mattei, Formes intégrables holomorphes singulières,
Astérisque, 97 (1982), 193pp. |
| [7] |
D. Cerveau and R. Moussu,
Groupes d'automorphismes de $ ({\mathbb{C}},0)$ et équations différentielles $ ydy+··· = 0$, Bull. Soc. Math. France, 116 (1988), 459-488.
doi: 10.24033/bsmf.2108. |
| [8] |
D. Cerveau and J. Mozo Fernández,
Classification analytique des feuilletages singuliers réduits de codimension 1 en dimension $ n≥3$, Erg. Th. and Dyn. Systems, 22 (2002), 1041-1060.
doi: 10.1017/S0143385702000561. |
| [9] |
V. Cossart, Desingularization in dimension 2, in Resolution of Surface Singularities,
Lecture Notes in Mathematics, Springer-Verlag, 1101 (1984), 79-98. |
| [10] |
P. Fernández Sánchez and J. Mozo Fernández,
Quasi-ordinary cuspidal foliations in $ ({\mathbb{C}}^3,0)$, J. Differential Equations, 226 (2006), 250-268.
doi: 10.1016/j.jde.2005.09.006. |
| [11] |
P. Fernández Sánchez, J. Mozo Fernández and H. Neciosup,
On codimension one foliations with prescribed cuspidal separatrix, J. Differential Equations, 256 (2014), 1702-1717.
doi: 10.1016/j.jde.2013.12.002. |
| [12] |
J. Giraud, Desingularization in low dimension, in Resolution of Surface Singularities, Lecture Notes in Mathematics, Springer-Verlag, 1101 (1984), 50-78. |
| [13] |
J. P. Jouanolou,
Équations de Pfaff Algébriques, Lecture Notes in Mathematics, 708, Springer-Verlag, 1979. |
| [14] |
F. Loray,
A preparation theorem for codimension one foliations, Ann. of Math., 163 (2006), 709-722.
doi: 10.4007/annals.2006.163.709. |
| [15] |
B. Malgrange,
Frobenius avec singularités. I. Codimension un, Inst. Hautes Études Sci. Publ. Math., 46 (1976), 163-173.
|
| [16] |
J. F. Mattei and R. Moussu,
Holonomie et intégrales premiéres, Ann. Sci. École Normale Sup., 13 (1980), 469-523.
|
| [17] |
R. Meziani,
Classification analytique d'équations différentielles $ ydy+··· = 0$ et espaces de modules, Bol. Soc. Brasil Mat., 27 (1996), 23-53.
doi: 10.1007/BF01246703. |
| [18] |
R. Meziani and P. Sad,
Singularités nilpotentes et intégrales premières, Publ. Mat., 51 (2007), 143-161.
doi: 10.5565/PUBLMAT_51107_07. |
| [19] |
J. J. Morales-Ruiz,
Differential Galois Theory and Non-Integrability of Hamiltonian Systems, Birkhäuser Progress in Mathematics, 1999.
doi: 10.1007/978-3-0348-8718-2. |
| [20] |
Holonomie évanescente des équations différentielles dégénérées transverses, in Singularities and Dynamical Systems, North-Holland, 103 (1985), 161-173.
doi: 10.1016/S0304-0208(08)72123-6. |
show all references
References:
| [1] |
C. Camacho,
Dicritical singularities of holomorphic vector fields, Contemporary Mathematics, 269 (2001), 39-45.
doi: 10.1090/conm/269/04328. |
| [2] |
C. Camacho and A. Lins Neto,
Teoria Geométrica das Folheações, IMPA, Projeto Euclides, 1979. |
| [3] |
F. Cano,
Reduction of the singularities of codimension one holomorphic foliations in dimension three, Annals of Math, 160 (2004), 907-1011.
doi: 10.4007/annals.2004.160.907. |
| [4] |
F. Cano, M. Ravara-Vago and M. Soares,
Local Brunella's alternative I. RICH foliations, Int. Math. Res. Not. IMRN, 9 (2015), 2525-2575.
doi: 10.1093/imrn/rnu011. |
| [5] |
F. Cano and D. Cerveau,
Desingularization of non-dicritical holomorphic foliations and existence of separatrices, Acta Math., 169 (1992), 1-103.
doi: 10.1007/BF02392757. |
| [6] |
D. Cerveau and J. -F. Mattei, Formes intégrables holomorphes singulières,
Astérisque, 97 (1982), 193pp. |
| [7] |
D. Cerveau and R. Moussu,
Groupes d'automorphismes de $ ({\mathbb{C}},0)$ et équations différentielles $ ydy+··· = 0$, Bull. Soc. Math. France, 116 (1988), 459-488.
doi: 10.24033/bsmf.2108. |
| [8] |
D. Cerveau and J. Mozo Fernández,
Classification analytique des feuilletages singuliers réduits de codimension 1 en dimension $ n≥3$, Erg. Th. and Dyn. Systems, 22 (2002), 1041-1060.
doi: 10.1017/S0143385702000561. |
| [9] |
V. Cossart, Desingularization in dimension 2, in Resolution of Surface Singularities,
Lecture Notes in Mathematics, Springer-Verlag, 1101 (1984), 79-98. |
| [10] |
P. Fernández Sánchez and J. Mozo Fernández,
Quasi-ordinary cuspidal foliations in $ ({\mathbb{C}}^3,0)$, J. Differential Equations, 226 (2006), 250-268.
doi: 10.1016/j.jde.2005.09.006. |
| [11] |
P. Fernández Sánchez, J. Mozo Fernández and H. Neciosup,
On codimension one foliations with prescribed cuspidal separatrix, J. Differential Equations, 256 (2014), 1702-1717.
doi: 10.1016/j.jde.2013.12.002. |
| [12] |
J. Giraud, Desingularization in low dimension, in Resolution of Surface Singularities, Lecture Notes in Mathematics, Springer-Verlag, 1101 (1984), 50-78. |
| [13] |
J. P. Jouanolou,
Équations de Pfaff Algébriques, Lecture Notes in Mathematics, 708, Springer-Verlag, 1979. |
| [14] |
F. Loray,
A preparation theorem for codimension one foliations, Ann. of Math., 163 (2006), 709-722.
doi: 10.4007/annals.2006.163.709. |
| [15] |
B. Malgrange,
Frobenius avec singularités. I. Codimension un, Inst. Hautes Études Sci. Publ. Math., 46 (1976), 163-173.
|
| [16] |
J. F. Mattei and R. Moussu,
Holonomie et intégrales premiéres, Ann. Sci. École Normale Sup., 13 (1980), 469-523.
|
| [17] |
R. Meziani,
Classification analytique d'équations différentielles $ ydy+··· = 0$ et espaces de modules, Bol. Soc. Brasil Mat., 27 (1996), 23-53.
doi: 10.1007/BF01246703. |
| [18] |
R. Meziani and P. Sad,
Singularités nilpotentes et intégrales premières, Publ. Mat., 51 (2007), 143-161.
doi: 10.5565/PUBLMAT_51107_07. |
| [19] |
J. J. Morales-Ruiz,
Differential Galois Theory and Non-Integrability of Hamiltonian Systems, Birkhäuser Progress in Mathematics, 1999.
doi: 10.1007/978-3-0348-8718-2. |
| [20] |
Holonomie évanescente des équations différentielles dégénérées transverses, in Singularities and Dynamical Systems, North-Holland, 103 (1985), 161-173.
doi: 10.1016/S0304-0208(08)72123-6. |
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