Rational values of transcendental functions and arithmetic dynamics

We count algebraic points of bounded height and degree on the graphs of certain functions analytic on the unit disk, obtaining a bound which is polynomial in the degree and in the logarithm of the multiplicative height. We combine this work with p-adic methods to obtain, for each positive ", an upper bound of the form cD3n=4+"n on the number of irreducible factors of Pºn(X) —Pºn(α) over K, where K is a number field, P is a polynomial of degree D ≥ 2 over K, Pºn is the n-th iterate of P, α is a point in K for which { Pºn(α) : n ∈ N} is infinite and c depends effectively on P, α, [K : Q] and ε.