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

Abstract

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.