Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-25T07:25:56.987Z Has data issue: false hasContentIssue false
  • English
  • Français

Permutable functions concerning differential equations

Part of: Complex dynamical systems Entire and meromorphic functions, and related topics

Published online by Cambridge University Press:  09 April 2009

X. Hua ,
R. Vaillancourt  and
X. L. Wang
X. Hua
Affiliation:
Department of Mathematics and StatisticsUniversity of OttawaOttawa, ON, K1N [email protected]@ottawa.ca
R. Vaillancourt
Affiliation:
Department of Mathematics and StatisticsUniversity of OttawaOttawa, ON, K1N [email protected]@ottawa.ca
X. L. Wang
Affiliation:
Department of Applied MathematicsNanjing University of Financeand EconomicsNanjing 210003, [email protected]
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.

Let f and g be two permutable transcendental entire functions. Assume that f is a solution of a linear differential equation with polynomial coefficients. We prove that, under some restrictions on the coefficients and the growth of f and g, there exist two non-constant rational functions R1 and R2 such that R1 (f) = R(g). As a corollary, we show that f and g have the same Julia set: J(f) = J(g). As an application, we study a function f which is a combination of exponential functions with polynomial coefficients. This research addresses an open question due to Baker.

MSC classification

Secondary: 30D05: Functional equations in the complex domain, iteration and composition of analytic functions 37F10: Polynomials; rational maps; entire and meromorphic functions 37F50: Small divisors, rotation domains and linearization; Fatou and Julia sets
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 83 , Issue 3 , December 2007 , pp. 369 - 384
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Baker, I. N., ‘Wandering domains in the iteration of entire functions’. Proc. London Math. Soc. 3 (1984). 563–576.CrossRefGoogle Scholar
[2]Bergweiler, W. and Hinkkanen, A.. ‘On semiconjugation of entire functions’. Math. Proc. Cambridge Philos. Soc. 126 (1999). 565–574.CrossRefGoogle Scholar
[3]Chuang, C. T. and Yang, C. C., Fix-Points and Factorization of Meromorphic Functions (World Scientific. Singapore. 1990).Google Scholar
[4]Fatou, P.. ‘Sur les équations fonctionelles’. Bull. Soc. Math. France 47 (1919). 161–271.CrossRefGoogle Scholar
[5]Fatou, P.. ‘Sur les équations fonctionelles’. Bull. Soc. Math. France 48 (1920). 33–94.CrossRefGoogle Scholar
[6]Fatou, P.. ‘Sur les équations fonctionelles’, Bull. Soc. Math. France 48 (1920), 208–314.CrossRefGoogle Scholar
[7]Gross, F. and Osgood, C. F.. ‘On fixed points of composite entire functions’. J. London Math. Soc. 28 (1983). 57–61.CrossRefGoogle Scholar
[8]Hayman, W. K., Meromorphic functions (Clarendon Press. Oxford. 1964).Google Scholar
[9]Hua, X. H. and Yang, C. C., Dynamics of transcendental functions (Gordon and Breach Science Publishers. 1998).Google Scholar
[10]Julia, G., ‘Mémoire sur la permutabilité des fractions rationnelles’, Ann. Sci. École Norm. Sup. 39 (1922), 131–215.CrossRefGoogle Scholar
[11]Laine, I., Nevanlinna Theory and Complex Differential Equations (Walter de Gruyter. Berlin-New York. 1993).CrossRefGoogle Scholar
[12]Liao, L. W. and Yang, C. C., ‘Some further results on the Julia sets of two permutable entire functions’, Rocky Mountain J. Math. to appear.Google Scholar
[13]Morosawa, S., Nishimura, Y.. Taniguchi, M. and Ueda, T., Holomorphic Dynamics (Cambridge University Press, 2000).Google Scholar
[14]Ng, T. W.. ‘Permutable entire functions and their Julia sets’, Math. Proc. Cambridge Philos. Soc. 131(2001). 129–138.CrossRefGoogle Scholar
[15]Polya, G. and Szegö, G.. Problems and Theorems in Analysis I (Springer. New York. 1972).Google Scholar
[16]Poon, K. K. and Yang, C. C.. ‘Dynamical behavior of two permutable entire functions’. Ann. Polon. Math. 168(1998), 159–163.CrossRefGoogle Scholar
[17]Wang, X. L.. Hua, X. H.. Yang, C. C. and Yang, D. G., ‘Dynamics of permutable transcendental entire functions’. Rocky Mountain J. Math. 36 (2006), 2041–2055.CrossRefGoogle Scholar
[18]Wang, X. L. and Yang, C. C.. ‘On the Fatou components of two permutable transcendental entire functions’. J Math. Anal. Appl. 278 (2003). 512–526.CrossRefGoogle Scholar
[19]Zheng, J. H. and Zhou, Z. Z.. ‘Permutability of entire functions satisfying certain differential equations’. Tohoku Math. J. 40 (1988). 323–330.CrossRefGoogle Scholar