Quasi-geodesics in the Cannon-Thurston metric

A closed fibered 3-manifold admits a complete hyperbolic metric if and only if it has a fibration with a pseudo-Anosov monodromy. The stable and the unstable laminations associated to the pseudo-Anosov homeomorphism on the fiber surface give rise to a natural metric on the 3-manifold, the Cannon-Thurston metric, which is quasi-isometric to the hyperbolic metric.
In this paper, we describe a specific family of quasi-geodesics in the Cannon-Thurston metric. We use the main results of this article in a companion paper to obtain statistics for typical geodesics with respect to various natural measures on the 2-sphere, thus giving a geometric criterion for singularity between some of these measure classes.