  In this paper we study the minimum dilatation pseudo-Anosov mapping classes
coming from fibrations over the circle of a single 3-manifold, the mapping
torus for the "simplest pseudo-Anosov braid". The dilatations that arise
include the minimum dilatations for orientable mapping classes for genus
g=2,3,4,5,8 as well as Lanneau and Thiffeault's conjectural minima for
orientable mapping classes, when g = 2,4 (mod 6). Our examples also show that
the minimum dilatation for orientable mapping classes is strictly greater than
the minimum dilatation for non-orientable ones when g = 4,6,8.
