In this talk I will present some results concerning the existence and properties of absorbing domains for holomorphic self-maps f of a hyperbolic domain U in C such that the iterates of f converge to a boundary point of U. In particular I will discuss the relation between the type of f (in the sense of Baker-Pommerenke-Cowen classification), its dynamical behavior and the simple-connectedness of absorbing domain. This is a joint work with K.Baranski, N.Fagella and X.Jarque.