Ramification in iterated towers for rational functions

Abstract

Let \({\phi(x)}\) be a rational function of degree > 1 defined over a number field K and let \({\Phi_{n}(x,t) = \phi^{(n)}(x)-t \in K(x,t)}\) where \({\phi^{(n)}(x)}\) is the nth iterate of \({\phi(x)}\). We give a formula for the discriminant of the numerator of Φn(x, t) and show that, if \({\phi(x)}\) is postcritically finite, for each specialization t0 of t to K, there exists a finite set \({S_{t_0}}\) of primes of K such that for all n, the primes dividing the discriminant are contained in \({S_{t_0}}\).