One-dimensional dynamics, intermediate regularity, and critical exponents

We will survey on several results concerning one-dimensional dynamics for group actions. After recalling the first results together with Deroin and Kleptsyn on (almost) sharp versions of Denjoy's theorem and Kopell's lemma for commuting diffeomorphisms, we will concentrate on new results. In particular, we will discuss the case of nilpotent group actions, with an emphasis on Farb-Franks' action (joint with Castro and Jorquera), as well as critical regularities for actions on the interval. Several open problems will be addressed.