  In this paper, we consider minimal equicontinuous actions of discrete
countably generated groups on Cantor sets, obtained from the arboreal
representations of absolute Galois groups of fields. In particular, we study
the asymptotic discriminant of these actions.
  The asymptotic discriminant is an invariant obtained by restricting the
action to a sequence of nested clopen sets, and studying the isotropies of the
enveloping group actions in such restricted systems. An enveloping (Ellis)
group of such an action is a profinite group.
  A large class of actions of profinite groups on Cantor sets is given by
arboreal representations of absolute Galois groups of fields. We show how to
associate to an arboreal representation an action of a discrete group, and give
examples of arboreal representations with stable and wild asymptotic
discriminant.
