Contracting rigid germs in higher dimensions

Matteo Ruggiero[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.

Annales de l’institut Fourier (2013)

  • Volume: 63, Issue: 5, page 1913-1950
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Ruggiero, Matteo. "Contracting rigid germs in higher dimensions." Annales de l’institut Fourier 63.5 (2013): 1913-1950. <http://eudml.org/doc/275489>.

@article{Ruggiero2013,
abstract = {Following Favre, we define a holomorphic germ $f:(\mathbb\{C\}^d, 0) \rightarrow (\mathbb\{C\}^d, 0)$ 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.},
affiliation = {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.},
author = {Ruggiero, Matteo},
journal = {Annales de l’institut Fourier},
keywords = {holomorphic fixed point germs; contracting rigid germs; normal forms; renormalization; resonances; critical set; holomorphic germs in dimension 3},
language = {eng},
number = {5},
pages = {1913-1950},
publisher = {Association des Annales de l’institut Fourier},
title = {Contracting rigid germs in higher dimensions},
url = {http://eudml.org/doc/275489},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Ruggiero, Matteo
TI - Contracting rigid germs in higher dimensions
JO - Annales de l’institut Fourier
PY - 2013
PB - Association des Annales de l’institut Fourier
VL - 63
IS - 5
SP - 1913
EP - 1950
AB - Following Favre, we define a holomorphic germ $f:(\mathbb{C}^d, 0) \rightarrow (\mathbb{C}^d, 0)$ 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.
LA - eng
KW - holomorphic fixed point germs; contracting rigid germs; normal forms; renormalization; resonances; critical set; holomorphic germs in dimension 3
UR - http://eudml.org/doc/275489
ER -

References

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  12. Jasmin Raissy, Torus actions in the normalization problem, J. Geom. Anal. 20 (2010), 472-524 Zbl1203.37083MR2579518
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  15. Shlomo Sternberg, Local contractions and a theorem of Poincaré, Amer. J. Math. 79 (1957), 809-824 Zbl0080.29902MR96853

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