  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 $f_\lambda$.
