  In this work, we study arithmetic dynamical questions for formal groups and
$p$-adic dynamical systems in higher dimensions, and thus we generalize some
earlier results that appeared in the literature. We prove that for any two
higher-dimensional formal groups over the ring of $p$-adic integers, if they
have infinitely many torsion points in common, then they coincide. We also
prove that if $\mathcal{D}_1$ and $\mathcal{D}_2$ are $d$-dimensional stable
$p$-adic dynamical systems with infinitely many preperiodic points in common,
then $\mathcal{D}_1$ and $\mathcal{D}_2$ coincide.
