Prime-powered images and irreducible polynomials in dynamical semigroups

Let $G=⟨x^d+c_1,\dots ,x^d+c_s⟩$ be a semigroup generated under composition for some $c_1,\dots ,c_s\in \mathbb{Z}$ and some $d\ge 2$. Then we prove that, outside of an exceptional one-parameter family, $G$ contains a large and explicit subset of irreducible polynomials if and only if it contains at least one irreducible polynomial. In particular, this conclusion holds when $G$ is generated by at least $s\ge 3$ polynomials when $d$ is odd and at least $s\ge 5$ polynomials when $d$ is even. To do this, we prove a classification result for prime powered iterates under $f(x)=x^d+c$ when $c\in \mathbb{Z}$ is nonzero. Namely, if $f^n(\alpha )=y^p$ for some $n\ge 4$, some $\alpha ,y\in \mathbb{Z}$, and some prime $p\mathrm{\mid }d$, then $\alpha$ and $y^p$ are necessarily preperiodic and periodic points for $f$ respectively. Moreover, we note that $n=4$ is the smallest possible iterate for which one may make this conclusion.