Simultaneously preperiodic points and colliding orbits for families of polynomials in characteristic $p$

Abstract

Let $k=\overline{\mathbb{F}}_p$ and let $f\in k[X]$ be a polynomial of degree at least $2$ such that $f-f(0)$ is non-additive. For $α,β$ in an algebraically closed extension $L$ of $k$, we prove that there are infinitely many $λ\in L$ for which both points are preperiodic under $f(X)+λ$ if and only if $α,β\in k$ or $f(β)-f(α)\in E_f$. Here $E_f=\{c\in G_f:(f-f(0))^n(c)\in G_f\text{ for every }n\ge0\}$, where $G_f=\{c\in k : f(X+c)-f(X)\in k\}$. We also provide a necessary and sufficient criterion for the corresponding colliding orbits set to be infinite.