Results on the postsingular set of compositions of transcendental entire functions

In this paper, we study the dynamics of commuting transcendental entire functions $f$ and $g$, where $g=a{f}^p+b$ with $a,b\in \mathbb{C}$, $p\in \mathbb{N}$, and $a\mathrm{\ne }0,1$. We examine how singular values and postsingular sets behave under composition. Within this framework, we show that if one of the functions is postsingularly finite (respectively, postsingularly bounded, hyperbolic), then the other function also has this property, and so do their compositions. As an application, we derive several results concerning transcendental semigroups, including situations in which Eremenko’s conjecture is satisfied.