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We consider semigroup actions on the unit interval generated by strictlyincreasing $C^r$-maps. We assume that one of the generators has a pair of fixedpoints, one attracting and one repelling, and a heteroclinic orbit thatconnects the repeller and attractor, and the other generators form a robustblender, which can bring the points from a small neighborhood of the attractorto an arbitrarily small neighborhood of the repeller. This is a model settingfor partially hyperbolic systems with one central direction.  We show that, under additional conditions on the non-linearity and theSchwarzian derivative, the above semigroups exhibit, $C^r$-generically for anyr, arbitrarily fast growth of the number of periodic points as a function ofthe period. We also show that a $C^r$-generic semigroup from the class underconsideration supports an ultimately complicated behavior called universaldynamics.

Degenerate behavior in non-hyperbolic semigroup actions on the interval: fast growth of periodic points and universal dynamics