Ritt's theorem and the Heins map in hyperbolic complex manifolds

Let X be a Kobayashi hyperbolic complex manifold, and assume that X does not contain compact complex submanifolds of positive dimension (e.g., X Stein). We shall prove the following generalization of Ritt's theorem: every holomorphic self-map f:X -> X such that f (X) is relatively compact in X has a unique fixed point tau(f) is an element of X, which is attracting. Furthermore, we shall prove that tau(f) depends holomorphically on f in a suitable sense, generalizing results by Heins, Joseph-Kwack and the second author.