We study a topologically transitive fibered system over a horseshoe map that is derived from a homoclinic class. In particular, this class contains saddles of different indices and hence is not hyperbolic. This class possesses a very rich fiber structure (uncountably many trivial and uncountably many non-trivial spines). Moreover, we observe that the spectrum of the central Lyapunov exponents contains a gap. The fibered system is naturally associated to an iterated function system that is genuinely non-contracting.