Arboreal Galois representations and uniformization of polynomial dynamics

Given a polynomial f of degree d defined over a complete local field, we construct a biholomorphic change of variables defined in a neighbourhood of infinity which transforms the action z->f(z) to the multiplicative action z->z^d. The relation between this construction and the Bottcher coordinate in complex polynomial dynamics is similar to the relation between the complex uniformization of elliptic curves, and Tate's p-adic uniformization. Specifically, this biholomorphism is Galois equivariant, reducing certain questions about the Galois theory of preimages by f to questions about multiplicative Kummer theory.