Finitely ramified iterated extensions

Let K be a number field, t a parameter, F = K(t), and φ(x)∈ K [x] a polynomial of degree d ≥ 2. The polynomial Φn(x,t) = φ^n (x) − t ∈ F[x], where φ^ n = φ ^ φ ^ … ^ φ is the n-fold iterate of φ, is irreducible over F; we give a formula for its discriminant. Let F be the field obtained by adjoining to F all roots (in a fixed ) of Φn(x,t) for all n ≥ 1; its Galois group Gal(Fφ/F) is the iterated monodromy group of φ. The iterated extension Fφ is finitely ramified over F if and only if φ is postcritically finite. We show that, moreover, for post-critically finite φ, every specialization of Fφ/F at t = t0 ∈ K is finitely ramified over K, pointing to the possibility of studying Galois groups of number fields with restricted ramification via tree representations associated to iterated monodromy groups of postcritically finite polynomials. We discuss the wildness of ramification in some of these representations, describe prime decomposition in terms of certain finite graphs, and also give some examples of monogène number fields that arise from the construction