Iterates of an analytic family of holomorphic mapping and fixed points on a product

Larbi Belkhchicha; Jean-Pierre Vigué

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2003)

  • Volume: 14, Issue: 1, page 23-31
  • ISSN: 1120-6330

Abstract

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In this paper, we consider an analytic family of holomorphic mappings f : M × X → X and the sequence f n of iterates of f . If the sequence is not compactly divergent, there exists an unique retraction ρ ( m , . ) adherent to the sequence f n ( m , . ) . If X is a strictly convex taut domain in C n and if the image Λ ( ρ ( m , . ) ) of ρ ( m , . ) is of dimension ≥ 1 , we prove that Λ ( ρ ( m , . ) ) does not depend from m ∈ M . We apply this result to the existence of fixed points of holomorphic mappings on the product of two bounded strictly convex domains.

How to cite

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Belkhchicha, Larbi, and Vigué, Jean-Pierre. "Itérées d’une famille analytique d’applications holomorphes et points fixes sur un produit." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 14.1 (2003): 23-31. <http://eudml.org/doc/252328>.

@article{Belkhchicha2003,
author = {Belkhchicha, Larbi, Vigué, Jean-Pierre},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Iterates of holomorphic mappings; Holomorphic retractions; Fixed points on a product; iterates of holomorphic mappings; holomorphic retractions; fixed points on a product},
language = {fre},
month = {3},
number = {1},
pages = {23-31},
publisher = {Accademia Nazionale dei Lincei},
title = {Itérées d’une famille analytique d’applications holomorphes et points fixes sur un produit},
url = {http://eudml.org/doc/252328},
volume = {14},
year = {2003},
}

TY - JOUR
AU - Belkhchicha, Larbi
AU - Vigué, Jean-Pierre
TI - Itérées d’une famille analytique d’applications holomorphes et points fixes sur un produit
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2003/3//
PB - Accademia Nazionale dei Lincei
VL - 14
IS - 1
SP - 23
EP - 31
LA - fre
KW - Iterates of holomorphic mappings; Holomorphic retractions; Fixed points on a product; iterates of holomorphic mappings; holomorphic retractions; fixed points on a product
UR - http://eudml.org/doc/252328
ER -

References

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  1. ABATE, M., Iteration theory of holomorphic maps on taut manifolds. Mediterranean Press, Rende, Cosenza1990. Zbl0747.32002MR1098711
  2. ABATE, M., Iteration theory, compactly divergent sequences and commuting holomorphic maps. Ann. Scuola Norm. Sup. Pisa Cl. Sci., (4), 18, 1991, 167-191. Zbl0760.32014MR1129300
  3. ABATE, M., Iterations of holomorphic families. Rend. Istit. Mat. Univ. Trieste, 26, 1994, n. 1-2, 141-150. Zbl0864.32019MR1363918
  4. BEDFORD, E., On the automorphism group of a Stein manifold. Math. Ann., 266, 1983, 215-227. Zbl0532.32014MR724738DOI10.1007/BF01458443
  5. CARTAN, H., Sur les fonctions de plusieurs variables complexes: l’itération des transformations intérieures d’un domaine borné. Math. Z., 35, 1932, 760-773. Zbl0004.40602MR1545327DOI10.1007/BF01186587
  6. CARTAN, H., Sur les rétractions d’une variété. C. R. Acad. Sc. Paris Série I Math., 303, 1986, 715-716. Zbl0609.32021MR870703

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