On hyperbolic once-punctured-torus bundles

For a hyperbolic once-punctured-torus bundle over a circle, a choice of normalization determines a family of arcs in the Riemann sphere. We show that, in each arc in the family, the set of cusps is dense and forms a single orbit of a finitely generated semigroup of Möbius transformations. This was previously known for the case of the complement of the figure-eight knot.