We consider a $C^1$-open class of step one skew products of
homeomorphisms defined in an interval. The class appeared in previous
works of Diaz and Rocha (as a result of a bifurcation), it can also be
obtained by perturbation of another class of systems, studied by Alseda
and Misiurewicz and by Fan, Simon, and Toth. I will present some results
on the structure of the simplex of invariant measures, as well as some
results on approximation of ergodic measures with periodic orbits or
with horseshoes. This is a joint work with Lorenzo Diaz and Katrin Gelfert.