On simultaneously preperiodic points for one-parameter families of polynomials in characteristic $p$

Abstract

For a field $L$ of characteristic $p$, a polynomial $f\in {\overline{\mathbb{F}}}_p[x]$ and $\hat{I}\pm ,{\hat{I}}^2\in L$, let $\mathrm{Prep}(f;\hat{I}\pm ,{\hat{I}}^2)$ be the set of all $\hat{I}»\in \overline{L}$ such that both $\hat{I}\pm$ and ${\hat{I}}^2$ are preperiodic under the action of $f_{\hat{I}}»(x):=f(x)+\hat{I}»$. Ghioca and Hsia proved that for certain families of polynomials, this set is infinite if and only if $f(\hat{I}\pm )=f({\hat{I}}^2)$ or $\hat{I}\pm ,{\hat{I}}^2\in {\overline{\mathbb{F}}}_p$. Building on their work, we determine when $\mathrm{Prep}(f;\hat{I}\pm ,{\hat{I}}^2)$ is infinite for most of the remaining binomial cases that were left open. Specifically, let $f(x)=c_1{x}^{d_1}+c_2{x}^{d_2}\in {\overline{\mathbb{F}}}_p[x]$, where $c_i\in {\overline{\mathbb{F}}}_p^{\ast }$, $1\le d_1<d_2$ and $d_i=p^{{\ell }_i}{s}_i$ with ${\ell }_i\ge 0$ and $p\nmid s_i$. We prove that if $p^{{\ell }_2}(s_2-1)<p^{{\ell }_1}(s_1-1)$, then $\mathrm{Prep}(f;\hat{I}\pm ,{\hat{I}}^2)$ is infinite if and only if $f(\hat{I}\pm )=f({\hat{I}}^2)$ or $\hat{I}\pm ,{\hat{I}}^2\in {\overline{\mathbb{F}}}_p$. The key idea of the proof is to use the parameters $\hat{I}{»}_{\overline{\hat{I}}\pm }:=\overline{\hat{I}}\pm -f(\overline{\hat{I}}\pm )$ associated to suitable elements $\overline{\hat{I}}\pm \in \overline{L}$ satisfying $f(\overline{\hat{I}}\pm )=f(\hat{I}\pm )$. As an application, we extend the work of Asgarli and Ghioca on the colliding orbits problem to binomials satisfying $s_2>1$ and $p^{{\ell }_2}(s_2-1)<p^{{\ell }_1}(s_1-1)$.

19 pages, main theorem improved, Section 5 added