On the Julia sets of two permutable entire functions (Q2496906)

Two entire functions \(f\) and \(g\) are called permutable if \(f \circ g=g \circ f.\) Fatou and Julia classified permutable polynomials. One intermediate result in their studies was that if \(f\) and \(g\) are permutable polynomials, then \(J(f)=J(g),\) where \(J(\cdot)\) denotes the Julia set. It is not known whether this last result holds for transcendental entire functions \(f\) and \(g\), nor is a classification of permutable entire functions known. Here the authors give several conditions under which they can determine the functions \(g\) which are permutable with a given function \(f\) and from which they then can deduce that \(J(f)=J(g).\) Here we only quote one result: Let \(f\) be a transcendental entire function such that \(f'\) has only one zero, \(f\) is pseudo-prime, \(f\) is not of the form \(H\circ Q\) where \(H\) is periodic and entire and \(Q\) is a nonlinear polynomial and \(f\) is not of the form \((z-a)^ne^{h(z)}+A\) where \(a\) is the zero of \(f'\), \(n\geq 2\) and \(h\) is a transcendental entire function with infinitely many zeros such that \(n+(z-a)h'(z)\) has no zeros. Then every nonlinear entire function \(g\) that is permutable with \(f\) has the form \(g(z)=cf^n(z)+b\) where \(n\) is a positive integer, \(c\) is a root of unity and \(b\) is a complex number. This implies that \(J(f)=J(g).\) The result complements a result of \textit{T. W. Ng} [Math. Proc. Camb. Philos. Soc. 131, 129--138 (2001; Zbl 0979.30015)] who obtained the same conclusion under a similar set of hypotheses, but who was assuming that \(f'\) has at least two zeros. Other results of this paper are variations and extensions of Ng's result.