There is a beautiful and well developed theory (due to Thurston, Fried and McMullen) classifying all of the fiberations over the circle of a given 3-manifold.
These fibrations have common characteristics in particular if one fibration has a monodromy that is pseudo-Anosov then all fibrations have this property, and the dilatations are related via an element in the group ring of the first homology of the manifold.
We will discuss a theory of fiberations of free-by-cyclic groups that was developed in analogy to the 3-manifold case.
In particular we will discuss Dowdall-Kapovich-Leininger's construction of an open cone of fiberations of a free-by-cyclic group, and their theorem that if the original outer automorphism was fully irreducible then the monodromy of each element in this cone is an irreducible train-track map.
We then describe a polynomial that packages all of the dilatations of all of these train-track maps (by joint work with Hironaka and Rafi).