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 $\alpha ,\beta$ in an algebraically closed extension $L$ of $k$, we prove that there are infinitely many $\lambda \in L$ for which both points are preperiodic under $f(X)+\lambda$ if and only if $\alpha ,\beta \in k$ or $f(\beta )-f(\alpha )\in E_f$. Here $E_f=\{c\in G_f:(f-f(0){)}^n(c)\in G_f\text{ for every }n\ge 0\}$, 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.

12 pages