In 2000, Kaloshin proved that C^r(r>1) generic diffeomorphisms in the Newhouse region have arbitrary growth of the number of periodic points. It is natural to ask whether generic partially hyperbolic diffeomorphisms with one-dimensional center bundle have the same property or not. In this talk, we investigate generic semigroup actions (or iterated function systems) on the interval as a toy model of partially hyperbolic systems. We will see that C^r(r>0) generic semigroup actions have arbitrary growth of the number of periodic points under some mild conditions. As a corollary, we obtain an open set of one-dimensional semi-groups in which the average growth of the number of periodic points is arbitrary large, but dynamics is very tame (it has unique attracting periodic point for any large iterations) along almost all path in the semigroup. If we have time enough, we also discuss genericity of `universal dynamics' under the mild conditions same as above.

This is a joint work with K.Shinohara and D.Turaev.