  We introduce the notion of a zebra structure on a surface, which is a more
general geometric structure than a translation structure or a dilation
structure that still gives a directional foliation of every slope. We are
concerned with the question of when a free homotopy class of loops (or a
homotopy class of arcs relative to endpoints) has a canonical representative or
family of representatives, either as closed leaves or chains of leaves joining
singularities. We prove that such representations exist if the surface has a
triangulation with edges joining singularities (in the zebra structure sense).
In the special case when the surface is closed, we describe several geometric
conditions that are equivalent to the existence of canonical representations in
every homotopy class of closed curves.
