  We introduce an anabelian approach to the study of arboreal Galois
representations and apply Tamagawa's anabelian version of the
N\'eron-Ogg-Shafarevich criterion to produce a dynamical analogue of this
criterion: unramified representations correspond to rational maps satisfying a
strong form of good reduction in terms of their critical locus. Subsequently,
we pursue a dynamical anlaogue of the N\'eron-Ogg-Shafarevich criterion in
terms of the more (dynamically) traditional arboreal representations, which
relates unramified arboreal representations to a certain separability condition
on the dynamical system. Finally, we relate the our criteria: the anabelian
criterion corresponds to the dynamical criterion as one varies the base point
around the critical locus. Along the way we develop effective criteria to
determine which primes are infinitely ramified in arboreal representations over
number fields, as well as the asymptotic growth of that ramification; we
conclude with examples and applications, especially to dynamical systems over
number fields.
