Permutable entire functions and their Julia sets (Q2743680)

In the 1920s, P. Fatou and G. Julia used their iteration theory to classify permutable polynomials; that is, to characterize polynomials \(f\) and \(g\) such that \(f\circ g=g\circ f\). They showed that if \(f\) and \(g\) are nonlinear and do not have a common iterate, then they are conjugate to monomials or to Chebychev polynomials. In the course of their proof they showed in particular that \(f\) and \(g\) have the same Julia set. Soon afterwards, J. F. Ritt extended the classification to rational functions. Here additional examples arising from the multiplication theorems of elliptic functions have to be taken into account. While Ritt did not use iteration, a proof of his result based on iteration was given by A.E. Eremenko. A corresponding classification of permutable entire functions is not known. There are, however, a number of results assuming that \(f\) and \(g\) are of a special form, and also some results showing that the Julia sets are equal under certain hypotheses. (All these results are cited.) In this interesting paper it is shown that if \(f\) satisfies certain hypotheses, then the only functions \(g\) commuting with \(f\) are of the form \(g(z)=af^{n}(z)+b\), where \(a\) is a root of unity, \(n\) a positive integer and \(b\in {\mathbb{C}}\). This implies that the Julia sets are equal. The hypotheses are: NEWLINENEWLINENEWLINE(i) \(f\) is not of the form \(H\circ Q\), where \(H\) is periodic and \(Q\) a polynomial. NEWLINENEWLINENEWLINE(ii) \(f\) is left-prime in the entire sense; that is, if \(f=\alpha \circ \beta\) with entire functions \(\alpha\) and \(\beta\), where \(\beta\) is transcendental, then \(\alpha\) is linear. NEWLINENEWLINENEWLINE(iii) \(f'\) has at least two distinct zeros. NEWLINENEWLINENEWLINE(iv) There exists \(N\in {\mathbb{N}}\) such that for any \(c\in {\mathbb{C}}\), the equations \(f(z)=c\) and \(f'(z)=0\) have at most \(N\) common solutions. NEWLINENEWLINENEWLINE(v) The orders of the zeros of \(f'\) are bounded by some \(M\in {\mathbb{N}}\). NEWLINENEWLINENEWLINEEven though this looks like a long list of hypotheses, the result appears to be the strongest one of its type. It is also shown that ``most'' entire functions satisfy the hypotheses. Specific examples are given by \(e^{z}+p(z)\) and \(\sin z+p(z)\), where \(p\) is a non-constant polynomial. One of the main tools used is a result on common right factors obtained from work of \textit{A. Eremenko} and \textit{L. A. Rubel} [Adv. Math. 124, No. 2, 334-354 (1996; Zbl 0871.30029)].