A Galois Criterion for Good Reduction via Orbital Arboreal Representations

Abstract: Let K be a non-archimedean local field and phi: P^1 -> P^1 a rational endomorphism of degree d >= 2 defined over K. We relate the reduction of phi to the Galois action on its preimage towers by introducing the orbital colimit G_{O+(x)} := colim_n Im(rho_{phi^n(x)}), a canonical Galois object encoding the dynamics along a forward orbit. Our main theorem (Theorem 3.1) gives a precise equivalence: phi has strict good reduction if and only if there exists a nonempty open set of integral points x in P^1(O_K) whose reductions avoid the reduced postcritical set PC(tilde{phi}), such that the reduced level-1 fibers have degree d and are etale -- equivalently, the associated fiber polynomials have unit leading coefficient and unit discriminant. In particular, for every such x, the extensions K(X_n(x))/K are unramified for all n >= 1. This yields a dynamical Neron--Ogg--Shafarevich criterion in arboreal terms, valid on the natural open locus. We provide complete proofs and explicit examples over Q_p.