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Permutable functions concerning differential equations

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

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 6N5Canadahua@mathstat.uottawa.caremi@ottawa.ca
R. Vaillancourt
Affiliation:
Department of Mathematics and StatisticsUniversity of OttawaOttawa, ON, K1N 6N5Canadahua@mathstat.uottawa.caremi@ottawa.ca
X. L. Wang
Affiliation:
Department of Applied MathematicsNanjing University of Financeand EconomicsNanjing 210003, JiangsuChinawangxiaoling@vip.163.com
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Abstract

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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

Information

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

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