Hostname: page-component-f554764f5-fr72s Total loading time: 0 Render date: 2025-04-08T16:21:59.367Z Has data issue: false hasContentIssue false
  • English
  • Français

Fixed Points of Commuting Holomorphic Maps Without Boundary Regularity

Published online by Cambridge University Press:  20 November 2018

Filippo Bracci
Filippo Bracci*
Affiliation:
Dipartimento di Matematica Pura ed Applicata, Università degli Studi di Padova, Via Belzoni 7, 35131 Padova, Italia, email: fbracci@math.unipd.it
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We identify a class of domains of ${{\mathbb{C}}^{n}}$ in which any two commuting holomorphic self-maps have a common fixed point.

Keywords

32A1032A4032H1532A30Holomorphic self-mapscommuting functionsfixed pointsWolff pointJulia’s Lemma
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 43 , Issue 3 , 01 September 2000 , pp. 294 - 303
Copyright
Copyright © Canadian Mathematical Society 2000

References

[1] Abate, M., Iteration theory of holomorphic maps on taut manifolds. Mediterranean Press, Rende, Cosenza 1989.Google Scholar
[2] Abate, M., The Lindelöf principle and the angular derivative in strongly convex domains. J. Analyse Math. 54 (1990), 189–228.Google Scholar
[3] Abate, M., Iteration theory, compactly divergent sequences and commuting holomorphic maps. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) (2) 18 (1991), 167–191.Google Scholar
[4] Abate, M. and Vigué, J. P., Common fixed points in hyperbolic Riemann surfaces and convex domains. Proc. Amer.Math. Soc. 112 (1991), 503–512.Google Scholar
[5] Behan, D. F., Commuting analytic functions without fixed points. Proc. Amer.Math. Soc. 37 (1973), 114–120.Google Scholar
[6] Bracci, F., Common fixed points of commuting holomorphic maps in the unit ball of Cn . Proc. Amer.Math. Soc. 127 (1999), 1133–1141.Google Scholar
[7] Bracci, F., Commuting holomorphic maps in strongly convex domains. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 27 (1998), 131–144.Google Scholar
[8] Cima, J. A. and Krantz, S. G., The Lindelöf principle and normal functions of several complex variables. Duke Math. J. 50 (1983), 303–328.Google Scholar
[9] Čirca, E. M., The theorems of Lindelöf and Fatou in Cn . Math. USSR-Sb. (4) 21 (1973), 619–639.Google Scholar
[10] de Fabritiis, C., Commuting holomorphic functions and hyperbolic automorphisms. Proc. Amer.Math. Soc. 124 (1996), 3027–3037.Google Scholar
[11] de Fabritiis, C. and Gentili, G., On holomorphic maps which commute with hyperbolic automorphisms. Adv. in Math. (2) 144 (1999), 119–136.Google Scholar
[12] Denjoy, A., Sur l’ itération des fonctions analytiques. C. R. Acad. Sci. Paris 182 (1926), 255–257.Google Scholar
[13] Frostman, O., Sur les produits de Blaschke. Kungl. Fysiogr. Sälsk. i Lund. Försh. 12 (1942), 169–182.Google Scholar
[14] Heins, M. H., A generalization of the Aumann-Carathodory “Starrheitssatz”. DukeMath. J. 8 (1941), 312–316.Google Scholar
[15] Huang, X., A non-degeneracy property of extremal mappings and iterates of holomorphic self-mappings. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) (3) 21 (1994), 399–419.Google Scholar
[16] Jarnicki, M. and Pflug, P., Invariant distances and metrics in complex analysis. W. de Gruyter, Berlin, New York, 1993.Google Scholar
[17] Julia, G., Extension nouvelle d’un lemme de Schwarz. ActaMath. 42 (1920), 349–355.Google Scholar
[18] Ma, D., On iterates of holomorphic maps. Math. Z. 207 (1991), 417–428.Google Scholar
[19] MacCluer, B. D., Iterates of holomorphic self-maps of the unit ball in CN. Michigan Math. J. 30 (1983), 97–106.Google Scholar
[20] Mercer, P. R., Complex geodesics and iterates of holomorphic maps on convex domains in Cn . Trans. Amer. Math. Soc. (1) 338 (1993), 201–211.Google Scholar
[21] Rudin, W., Function theory in the Unit Ball of Cn . Springer, Berlin, 1980.Google Scholar
[22] Shields, A. L., On fixed points of commuting analytic functions. Proc. Amer.Math. Soc. 15 (1964), 703–706.Google Scholar
[23] Stein, E. M., Boundary behaviour of holomorphic functions of several complex variables. Princeton University Press, Princeton, 1972.Google Scholar
[24] Wolff, J., Sur l’itération des fonctions bornées. C. R. Acad. Sci. Paris (1926), 200–201.Google Scholar
[25] Zhang, W. and Ren, F., Dynamics on weakly pseudoconvex domains. Chinese Ann.Math. Ser. B 4 (1995), 467–476.Google Scholar