Simultaneously preperiodic points for a family of polynomials in positive characteristic

In the goundbreaking paper [BD11] (which opened a wide avenue of research regarding unlikely intersections in arithmetic dynamics), Baker and DeMarco prove that for the family of polynomials $f_\lambda(x):=x^d+\lambda$ (parameterized by $\lambda\in\mathbb{C}$), given two starting points $a$ and $b$ in $\mathbb{C}$, if there exist infinitely many $\lambda\in\mathbb{C}$ such that both $a$ and $b$ are preperiodic under the action of $f_\lambda$, then $a^d=b^d$. In this paper we study the same question, this time working in a field of characteristic $p>0$. The answer in positive characteristic is more nuanced, as there are three distinct cases: (i) both starting points $a$ and $b$ live in $\Fpbar$; (ii) $d$ is a power of $p$; and (iii) not both $a$ and $b$ live in $\Fpbar$, while $d$ is not a power of $p$. Only in case~(iii), one derives the same conclusion as in characteristic $0$, i.e., that $a^d=b^d$. In case~(i), one has that for each $\lambda\in\Fpbar$, both $a$ and $b$ are preperiodic under the action of $f_\lambda$, while in case~(ii), one obtains that \emph{also} whenever $a-b\in\Fpbar$, then for each parameter $\lambda$, we have that $a$ is preperiodic under the action of $f_\lambda$ if and only if $b$ is preperiodic under the action of $f_\lambda$.