Following Favre, we define a holomorphic germ 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 and its linear action on the fundamental group of the complement of the critical set.
En suivant Favre, on dit qu’un germe holomorphe 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 et son action linéaire sur le groupe fondamental du complémentaire de l’ensemble critique.
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
@article{AIF_2013__63_5_1913_0, author = {Ruggiero, Matteo}, title = {Contracting rigid germs in higher dimensions}, journal = {Annales de l'Institut Fourier}, pages = {1913--1950}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {63}, number = {5}, year = {2013}, doi = {10.5802/aif.2818}, mrnumber = {3186512}, zbl = {06284536}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2818/} }
TY - JOUR AU - Ruggiero, Matteo TI - Contracting rigid germs in higher dimensions JO - Annales de l'Institut Fourier PY - 2013 SP - 1913 EP - 1950 VL - 63 IS - 5 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2818/ DO - 10.5802/aif.2818 LA - en ID - AIF_2013__63_5_1913_0 ER -
%0 Journal Article %A Ruggiero, Matteo %T Contracting rigid germs in higher dimensions %J Annales de l'Institut Fourier %D 2013 %P 1913-1950 %V 63 %N 5 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2818/ %R 10.5802/aif.2818 %G en %F AIF_2013__63_5_1913_0
Ruggiero, Matteo. Contracting rigid germs in higher dimensions. Annales de l'Institut Fourier, Volume 63 (2013) no. 5, pp. 1913-1950. doi : 10.5802/aif.2818. https://aif.centre-mersenne.org/articles/10.5802/aif.2818/
[1] 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] Formal normal forms for holomorphic maps tangent to the identity, Discrete Contin. Dyn. Syst., (suppl.) (2005), pp. 1-10 | MR | Zbl
[3] 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] Structure des surfaces de Kato, Mém. Soc. Math. France (N.S.), 14, 1984 | Numdam | MR | Zbl
[5] Sur la classification des germes d’applications holomorphes contractantes, Math. Ann., Volume 280 (1988) no. 4, pp. 649-661 | DOI | MR | Zbl
[6] Class surfaces with curves, Tohoku Math. J. (2), Volume 55 (2003) no. 2, pp. 283-309 | DOI | MR | Zbl
[7] 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] Eigenvaluations, Ann. Sci. École Norm. Sup. (4), Volume 40 (2007) no. 2, pp. 309-349 | Numdam | MR | Zbl
[9] 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] On compact complex -folds with lines, Japan. J. Math. (N.S.), Volume 11 (1985) no. 1, pp. 1-58 | MR | Zbl
[11] Compact complex threefolds of class associated to polynomial automorphisms of , Publ. Mat., Volume 50 (2006) no. 2, pp. 401-411 | DOI | MR | Zbl
[12] Torus actions in the normalization problem, J. Geom. Anal., Volume 20 (2010) no. 2, pp. 472-524 | DOI | MR | Zbl
[13] Holomorphic maps from to , Trans. Amer. Math. Soc., Volume 210 (1988) no. 1, pp. 47-86 | MR | Zbl
[14] Rigidification of holomorphic germs with non-invertible differential, Michigan Math. J., Volume 60 (2011)
[15] Local contractions and a theorem of Poincaré, Amer. J. Math., Volume 79 (1957), pp. 809-824 | DOI | MR | Zbl
Cited by Sources: