  We consider a dynamical system consisting of a pair of commuting power
series, one noninvertible and another nontorsion invertible, of height one with
coefficients in the $p$-adic integers. Assuming that each point of the
dynamical system generates a Galois extension over the base field, we show that
these extensions are in fact abelian, and, using results and considerations
from the theory of the field of norms, we also show that the dynamical system
must include a torsion series of maximal order. From an earlier result, this
shows that the series must in fact be endomorphisms of some height-one formal
group.
