Pseudo-iteration semigroups and commuting holomorphic maps

Graziano Gentili; Fabio Vlacci

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1994)

  • Volume: 5, Issue: 1, page 33-42
  • ISSN: 1120-6330

Abstract

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A connection between iteration theory and the study of sets of commuting holomorphic maps is investigated, in the unit disc of C . In particular, given two holomorphic maps f and g of the unit disc into itself, it is proved that if g belongs to the pseudo-iteration semigroup of f (in the sense of Cowen) then - under certain conditions on the behaviour of their iterates - the maps f and g commute.

How to cite

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Gentili, Graziano, and Vlacci, Fabio. "Pseudo-iteration semigroups and commuting holomorphic maps." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 5.1 (1994): 33-42. <http://eudml.org/doc/244278>.

@article{Gentili1994,
abstract = {A connection between iteration theory and the study of sets of commuting holomorphic maps is investigated, in the unit disc of \( C \). In particular, given two holomorphic maps \( f \) and \( g \) of the unit disc into itself, it is proved that if \( g \) belongs to the pseudo-iteration semigroup of \( f \) (in the sense of Cowen) then - under certain conditions on the behaviour of their iterates - the maps \( f \) and \( g \) commute.},
author = {Gentili, Graziano, Vlacci, Fabio},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Iteration theory; Commuting holomorphic maps; Fixed points; commuting functions; Denjoy-Wolff theorem; iteration; pseudo-iteration semigroup},
language = {eng},
month = {3},
number = {1},
pages = {33-42},
publisher = {Accademia Nazionale dei Lincei},
title = {Pseudo-iteration semigroups and commuting holomorphic maps},
url = {http://eudml.org/doc/244278},
volume = {5},
year = {1994},
}

TY - JOUR
AU - Gentili, Graziano
AU - Vlacci, Fabio
TI - Pseudo-iteration semigroups and commuting holomorphic maps
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1994/3//
PB - Accademia Nazionale dei Lincei
VL - 5
IS - 1
SP - 33
EP - 42
AB - A connection between iteration theory and the study of sets of commuting holomorphic maps is investigated, in the unit disc of \( C \). In particular, given two holomorphic maps \( f \) and \( g \) of the unit disc into itself, it is proved that if \( g \) belongs to the pseudo-iteration semigroup of \( f \) (in the sense of Cowen) then - under certain conditions on the behaviour of their iterates - the maps \( f \) and \( g \) commute.
LA - eng
KW - Iteration theory; Commuting holomorphic maps; Fixed points; commuting functions; Denjoy-Wolff theorem; iteration; pseudo-iteration semigroup
UR - http://eudml.org/doc/244278
ER -

References

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  1. ABATE, M., Iteration theory of holomorphic maps on taut manifolds. Mediterranean Press, 1989. Zbl0747.32002MR1098711
  2. BAKER, I. N., Zusammensetzungen ganzer Funktionen. Math. Z., 69, 1958, 121-163. Zbl0178.07502MR97532
  3. BEHAN, D. F., Commuting analytic functions without fixed points. Proc. Amer. Math. Soc., 37, 1973, 114-120. Zbl0251.30009MR308378
  4. COWEN, C. C., Iteration and the solution of functional equations for functions analytic in the unit disk. Trans. Amer. Math. Soc., 265, 1981, 69-95. Zbl0476.30017MR607108DOI10.2307/1998482
  5. COWEN, C. C., Commuting analytic functions. Trans. Amer. Math. Soc., 283, 1981, 685-695. Zbl0542.30030MR737892DOI10.2307/1999154
  6. HADAMARD, J., Two works on iteration and related questions. Bull. Amer. Math. Soc, 50, 1944, 67-75. Zbl0061.26503MR9691
  7. KOENIGS, G., Recherches sur les substitutions uniformes. Bull. Sci. Math., 7, 1883, 340-357. JFM15.0114.01
  8. POMMERENKE, CH., On the iteration of analytic functions in half plane, I. J. Lond. Math. Soc., 19, 1979, 439-447. Zbl0398.30014MR540058DOI10.1112/jlms/s2-19.3.439
  9. SCHROEDER, E., Über unendlich viele Algorithmen zur Auflösung der Gleichungen. Math. Ann., 2, 1870, 317-363. JFM02.0042.02
  10. VALIRON, G., Fonctions analytiques. Presses Universitaires de France, Paris1954. Zbl0055.06702MR61658
  11. VESENTINI, E., Capitoli scelti della teoria delle funzioni olomorfe. Unione Matematica Italiana, 1984. Zbl0691.30003
  12. WOLFF, J., Sur une généralisation d'un théorème de Schwarz. C.R. Acad. Sci. Paris, 182, 1926, 918-920. JFM52.0309.05
  13. WOLFF, J., Sur une généralisation d'un théorème de Schwarz. C.R. Acad. Sci. Paris, 183, 1926, 500-502. JFM52.0309.06

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