We show that in some topological classes of unimodal maps the measure- theoretical dynamical behaviour, as the existence of wild attractors or the rate of the convergence of a typical point to the measure-theoretical attractor, depends only on the order of the critical point. For infinitely renormalizable maps with bounded combinatorics, as Feigenbaum maps, we prove that the conjugacy between two maps in the same topological class and same critical order is always absolutely continuous. Co-author: C. G. Moreira (IMPA).