Generic automorphisms of handlebodies appear naturally in the study of isotopy classes of automophisms of orientable three-dimensional manifolds. Generic automorphisms remain as one of the least understood parts of this study. Given a generic automorphism of a handlebody one can construct a bidimensional lamination that is invariant under the automorphism. There is a growth rate associated to this lamination. It is known that, when this growth rate is minimal among all possible choices (an important property, for it measures the essential complexity of the automorphism), the lamination must have a certain incompressibility property. On this thesis we show that the process of