Dynamics of permutable transcendental entire functions (Q5906943)

Let \(f\) be a nonconstant meromorphic function, and denote by \(f^n\) the \(n\)-th iterate of \(f\). The Fatou set \(\mathcal{F}(f)\) is the maximal open set in which all iterates \(f^n\) are defined and form a normal family in the sense of Montel, while the complement of \(\mathcal{F}(f)\) in \(\widehat{\mathbb C}\) is the Julia set \(\mathcal{J}(f)\). Furthermore, denote by \(\text{sing }{f^{-1}}\) the set of finite singular values of \(f\) which are either critical values or asymptotic values. If this set is bounded, then \(f\) is said to be of bounded type. Finally, two nonconstant meromorphic functions are called permutable if \(f \circ g = g \circ f\). It is well-known that two permutable rational functions have the same Fatou and Julia set. A natural question is if this also holds for two permutable transcendental entire functions. In some special cases, this question was affirmatively solved by \textit{I. N. Baker} [Proc. Lond. Math. Soc. (3) 49, 563--576 (1984; Zbl 0523.30017)] and recently by K. K. Poon and C. C. Yang. Using a result of \textit{A. Eremenko} and \textit{M. Lyubich} [Ann. Inst. Fourier 42, 989--1020 (1992; Zbl 0735.58031)] the authors prove the following theorems. Theorem 1. Let \(f\) and \(g\) be two permutable transcendental entire functions of bounded type. Then \(\mathcal{J}(f)=\mathcal{J}(g)\). Theorem 2. Let \(f\) and \(g\) be two permutable transcendental entire functions. Then \(\mathcal{J}(f \circ g)=\mathcal{J}(f^n \circ g^m)\) for all positive integers \(m\) and \(n\).