The Farey tree and fibrations of the Whitehead link complement

Abstract

We show a curious connection between an example of dynamical forcing, where an existence of one periodic orbit for a dynamical system forces the existence of other periodic orbits, and the existence of different fibrations of a 3 dimensional manifold. Specifically, we establish a one to one correspondence between fibrations of the Whitehead link complement and the set of `simple orbit pairs' on the torus. The set of orbits is equipped with a complete order coming from a dynamical forcing relation, that corresponds to the Farey tree order of rational numbers. This corresponds to the order of the rational point on a fibered cone of the Whitehead link. For the proof we obtain an explicit description of all monodromies of the different fibrations of the Whitehead link complement.