Let $f(z) = 2z^3+z.$ What are the values of $z$ such that $z, f(z), f(f(z)), \dots$ is eventually periodic? I'm asking because I wish to classify all monic polynomials $p \in \mathbb{Z}[x]$ such that $p(z)|p(f(z)).$ If $S$ is the set of roots of $p,$ then we derive $r \in S \Rightarrow f(r) \in S,$ which eliminates the possibility of any roots that do not eventually lead to cycles.
Unfortunately, $0, f^{-1}(0) = \{0, \pm i/\sqrt{2}\}, f^{-2}(0), \dots$ all work. However, it would help to know that there are no other examples. Is this true?