  We study a class of chainable continua which contains, among others, all
inverse limit spaces generated by a single interval bonding map which is
piecewise monotone and locally eventually onto. Such spaces are realized as
attractors of non-hyperbolic surface homeomorphisms. Using dynamical properties
of the bonding map, we give conditions for existence of endpoints, characterize
the set of local inhomogeneities, and determine when it consists only of
endpoints. As a side product we also obtain a characterization of arcs as
inverse limits for piecewise monotone bonding maps, which is interesting in its
own right.
