Domains of explosion of the dynamic system generated by \(P(\lambda e^z+\mu e^{-z})\) (Q1377380)

The main result of the paper is the following. Let \(f(z)= P(\lambda e^z+ \mu e^{-z})\) where \(\lambda,\mu \in\mathbb{C}\), \(|\lambda |+| \mu| \neq 0\) and \(P\) is a non-constant polynomial. If for all \(s\in L=\{P(z): P'(z)=0\} \cup \{P(\pm 2\sqrt {\lambda\mu})\}\) we have \(\lim f^n(s)= \infty\) as \(n\to\infty\), then the Julia set of \(f\) is the whole plane. Some examples where \(P\) has the form \(az^n+b\), \(a,b,\in\mathbb{R}\), or \(P(z)= az^2+bz+c\) are discussed.