We consider semigroup actions on the unit interval generated by strictly
increasing $C^r$-maps. We assume that one of the generators has a pair of fixed
points, one attracting and one repelling, and a heteroclinic orbit that
connects the repeller and attractor, and the other generators form a robust
blender, which can bring the points from a small neighborhood of the attractor
to an arbitrarily small neighborhood of the repeller. This is a model setting
for partially hyperbolic systems with one central direction.
We show that, under additional conditions on the non-linearity and the
Schwarzian derivative, the above semigroups exhibit, $C^r$-generically for any
r, arbitrarily fast growth of the number of periodic points as a function of
the period. We also show that a $C^r$-generic semigroup from the class under
consideration supports an ultimately complicated behavior called universal
dynamics.