Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-12-01T10:43:40.776Z Has data issue: false hasContentIssue false
  • English
  • Français

Simultaneous Iteration by Entire or Rational Functions and Their Inverses

Part of: Entire and meromorphic functions, and related topics

Published online by Cambridge University Press:  09 April 2009

I. N. Baker  and
Zalman Rubinstein
I. N. Baker
Affiliation:
Department of Mathematics Imperial CollegeLondon S. W. 7, U. K.
Zalman Rubinstein
Affiliation:
Unversity of Michgan Ann Arbor, Machigan 48109 University of Haifa Haifa, Israel
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.

For a non-constant entire or rational function f normalized by f(0) = 0, f′(0) = 1, f″(0) ≠ 0, which is not a Möbius tranformation, the existence of a sequence is established which has the properties . The result certainly implies f(0)= |f(0)|= 1, so these conditions cannot be omitted. The condition f″ (0)≠ 0 can be replaced by f(k)(0) ≠ 0 for some k ≥ 2.

MSC classification

Secondary: 30D05: Functional equations in the complex domain, iteration and composition of analytic functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 34 , Issue 3 , June 1983 , pp. 364 - 367
Copyright
Copyright © Australian Mathematical Society 1983

References

[1]Baker, I. N., ‘The existence of fixpoints of entire functions’, Math. Z. 73 (1960), 280–284.CrossRefGoogle Scholar
[2]Baker, I. N., ‘Permutable power series and regular iteration’, J. Austral. Math. Soc. 2 (1962), 265–294.CrossRefGoogle Scholar
[3]Baker, I. N., ‘Fixpoints of polynomials and rational functions’, J. London Math. Soc. 39 (1964), 615–622.CrossRefGoogle Scholar
[4]Fatou, P., ‘Sur les équations fonctionelles’, Bull. Soc. Math. France 47 (1919), 161–271; 48 (1920), 33–94, 208–314.CrossRefGoogle Scholar
[5]Fatou, P., ‘Sur l'itération des fonctions transcendantes entières’, Acta Math. 47 (1926), 337–370.CrossRefGoogle Scholar
[6]Hayman, W. K., Meromorphic functions (Oxford University Press, 1964).Google Scholar
[7]Karamardian, Stepan, Ed., Fixed points algorithms and applications (Acadmic Press, 1977).Google Scholar
[8]Rosenbloom, P. C., ‘The fixpoints of entire functions’ Meddelanden Lunds Univ. Mat. Sem. Suppl. Band. M. Riesz (1952), 186–192.Google Scholar