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Given two monic polynomials f and g with coefficients in a number field K,and some a in K, we examine the action of the absolute Galois group of K on thedirected graph of iterated preimages of a under the correspondence g(y)=f(x),assuming that deg(f)>deg(g) and that gcd(deg(f), deg(g))=1. If a prime of Kexists at which f and g have integral coefficients, and at which a is notintegral, we show that this directed graph of preimages consists of finitelymany Galois-orbits. We obtain this result by establishing a p-adicuniformization of such correspondences, tenuously related to Bottcher'suniformization of polynomial dynamical systems over the complex numbers.

p-adic uniformization and the action of Galois on certain affine correspondences