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Galois Extensions of Height-One Commuting Dynamical Systems

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.