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 finding a lamination with such a property is algorithmic. On the other hand, we show that this said incompressibility property is not sufficient for the minimality of the growth rate. We propose a stronger property, which we called transverse tightness, and conjecture that it is a necessary condition for the growth rate to be minimal. We prove the conjecture in some particular cases. In addition to the results mentioned above, we develop methods to generate generic automorphisms of handlebodies, which we use to present some variety of examples.