  We strengthen the unpublished theorem of Gabai and Mosher that every depth
one sutured manifold contains a very full dynamic branched surface by showing
that the branched surface can be chosen to satisfy an additional property we
call veering. To this end we prove that every endperiodic map admits a periodic
splitting sequence of train tracks carrying its positive Handel-Miller
lamination. This completes step one of Gabai-Mosher's unpublished two-step
proof that every taut finite depth foliation of a compact, oriented, atoroidal
3-manifold is almost transverse to a pseudo-Anosov flow. Further, a veering
branched surface in a sutured manifold is a generalization of a veering
triangulation, and we extend some of the theory of veering triangulations to
this setting. In particular we show that the branched surfaces we construct are
unique up to a natural equivalence relation, and give an algorithmic way to
compute the foliation cones of Cantwell-Conlon.
