Contracting rigid germs in higher dimensions
[Germes rigides contractants en toute dimension]
Ruggiero, Matteo1
1 Fondation Mathématique Jacques Hadamard, Département de Mathématiques, UMR 8628 Université Paris-Sud 11-CNRS, Bâtiment 425, Faculté des Sciences d’Orsay, Université Paris-Sud 11, F-91405 Orsay Cedex, France. Centre de Mathématiques Laurent Schwartz, École Polytechnique, 91128 Palaiseau Cedex, France.
Annales de l'Institut Fourier, Tome 63 (2013) no. 5, pp. 1913-1950.

En suivant Favre, on dit qu’un germe holomorphe f:(ℂ d ,0)→(ℂ d ,0) est rigide si l’union de l’ensemble critique de tous ses itérés est à croisement normaux. Nous donnons une classification partielle des germes rigides contractants en toute dimension à conjugaison holomorphe près. On trouve des nouveaux phénomènes de résonance, entre la différentielle de f et son action linéaire sur le groupe fondamental du complémentaire de l’ensemble critique.

Following Favre, we define a holomorphic germ f:(ℂ d ,0)→(ℂ d ,0) to be rigid if the union of the critical set of all iterates has simple normal crossing singularities. We give a partial classification of contracting rigid germs in arbitrary dimensions up to holomorphic conjugacy. Interestingly enough, we find new resonance phenomena involving the differential of f and its linear action on the fundamental group of the complement of the critical set.

DOI : 10.5802/aif.2818
Classification : 37F25
Keywords: holomorphic fixed point germs, contracting rigid germs, normal forms, renormalization, resonances, critical set.
Mot clés : germes holomorphes, point fixe, germes rigides contractants, formes normales, renormalisation, résonances, ensemble critique.
Ruggiero, Matteo 1

1 Fondation Mathématique Jacques Hadamard, Département de Mathématiques, UMR 8628 Université Paris-Sud 11-CNRS, Bâtiment 425, Faculté des Sciences d’Orsay, Université Paris-Sud 11, F-91405 Orsay Cedex, France. Centre de Mathématiques Laurent Schwartz, École Polytechnique, 91128 Palaiseau Cedex, France. 
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Ruggiero, Matteo. Contracting rigid germs in higher dimensions. Annales de l'Institut Fourier, Tome 63 (2013) no. 5, pp. 1913-1950. doi : 10.5802/aif.2818. http://www.numdam.org/articles/10.5802/aif.2818/

[1] Abate, Marco; Raissy, Jasmin Formal Poincaré-Dulac renormalization for holomorphic germs to appear in Disc. Cont. Dyn. Syst. (2012). Preprint available at http://arxiv.org/abs/1008.0272

[2] Abate, Marco; Tovena, Francesca Formal normal forms for holomorphic maps tangent to the identity, Discrete Contin. Dyn. Syst., (suppl.) (2005), pp. 1-10 | MR | Zbl

[3] Berteloot, François Méthodes de changement d’échelles en analyse complexe, Ann. Fac. Sci. Toulouse Math. (6), Volume 15 (2006) no. 3, pp. 427-483 | DOI | Numdam | MR | Zbl

[4] Dloussky, Georges Structure des surfaces de Kato, Mém. Soc. Math. France (N.S.), 14, 1984 | Numdam | MR | Zbl

[5] Dloussky, Georges Sur la classification des germes d’applications holomorphes contractantes, Math. Ann., Volume 280 (1988) no. 4, pp. 649-661 | DOI | MR | Zbl

[6] Dloussky, Georges; Oeljeklaus, Karl; Toma, Matei Class VII 0 surfaces with b 2 curves, Tohoku Math. J. (2), Volume 55 (2003) no. 2, pp. 283-309 | DOI | MR | Zbl

[7] Favre, Charles Classification of 2-dimensional contracting rigid germs and Kato surfaces. I, J. Math. Pures Appl. (9), Volume 79 (2000) no. 5, pp. 475-514 | DOI | MR | Zbl

[8] Favre, Charles; Jonsson, Mattias Eigenvaluations, Ann. Sci. École Norm. Sup. (4), Volume 40 (2007) no. 2, pp. 309-349 | Numdam | MR | Zbl

[9] Kato, Masahide Compact complex manifolds containing “global” spherical shells. I, Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977), Kinokuniya Book Store, Tokyo, 1978, pp. 45-84 | MR | Zbl

[10] Kato, Masahide On compact complex 3-folds with lines, Japan. J. Math. (N.S.), Volume 11 (1985) no. 1, pp. 1-58 | MR | Zbl

[11] Oeljeklaus, Karl; Renaud, Julie Compact complex threefolds of class L associated to polynomial automorphisms of ℂ 3 , Publ. Mat., Volume 50 (2006) no. 2, pp. 401-411 | DOI | MR | Zbl

[12] Raissy, Jasmin Torus actions in the normalization problem, J. Geom. Anal., Volume 20 (2010) no. 2, pp. 472-524 | DOI | MR | Zbl

[13] Rosay, Jean-Pierre; Rudin, Walter Holomorphic maps from C n to C n , Trans. Amer. Math. Soc., Volume 210 (1988) no. 1, pp. 47-86 | MR | Zbl

[14] Ruggiero, Matteo Rigidification of holomorphic germs with non-invertible differential, Michigan Math. J., Volume 60 (2011)

[15] Sternberg, Shlomo Local contractions and a theorem of Poincaré, Amer. J. Math., Volume 79 (1957), pp. 809-824 | DOI | MR | Zbl

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