Finite ramification for preimage fields of postcritically finite morphisms

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Given a finite endomorphism φ of a variety X defined over the field of fractions K of a Dedekind domain, we study the extension K(φ⁻∞(α)): = ₙ ≥ ₁ K(φ⁻ⁿ(α)) generated by the preimages of α under all iterates of φ. In particular when φ is post-critically finite, i.e., there exists a non-empty, Zariski-open W X such that φ⁻¹(W) W and φ: W → X is étale, we prove that K(φ⁻∞(α)) is ramified over only finitely many primes of K. This provides a large supply of infinite extensions with restricted ramification, and generalizes results of Aitken-Hajir-Maire in the case X = A¹ and Cullinan-Hajir, Jones-Manes in the case X = P¹. Moreover, we conjecture that this finite ramification condition characterizes post-critically finite morphisms, and we give an entirely new result showing this for X = P¹. The proof relies on Faltings' theorem and a local argument.