| Generic automorphisms of handlebodies appear naturally in the classification of isotopy classes of automophisms of orientable and irreducible three-dimensional manifolds. Such automorphisms are in many ways analogous to pseudo-Anosov automorphisms of surfaces. One similarity has to do with the existence of invariant measured laminations: given a generic automorphism of a handlebody one can construct two laminations (one two-dimensional and one one-dimensional) which are invariant under a representative of the isotopy class of the automorphism. As for pseudo-Anosov automorphisms, there is a growth rate associated to these laminations. The laminations and their growth rates are not unique, so it is an important problem to determine when the growth rate is minimal. The invariant laminations with minimal growth rate capture the essential complexity of the isotopy class of the automorphism. It is known that when the growth rate is minimal the leaves of the lamination must satify a certain incompressibility condition.; In this thesis we address two main problems: that of building examples of generic automorphisms and that of characterizing minimal growth rate in terms of properties of the lamination.; On the problem of building automorphisms, we develop methods to generate examples sufficient to produce a range of interesting behaviors.; On the problem of characterizing minimal growth, we prove that the incompressibility condition is not sufficient for the growth to be minimal. We propose a stronger condition, which we called tightness, and conjecture that the tightness property of the laminations is equivalent to the minimality of the growth rate. We prove one direction: that tightness implies minimality of the growth rate. The other direction is proved under additional technical hypotheses. In particular, the conjecture is true if the genus of the handlebody is two. |
