Distortion elements in groups of interval (and circle) diffeomorphisms

An element $g$ of an abstract group $G$ is a distortion element if there exists a finite family $S$ in $G$ such that $g\in\langle S\rangle$ and the word-length of $g^n$ (w.r.t. $S$) grows sublinearly in $n$. This is a very general group theoretic notion, but does it have a dynamical interpretation when $G$ is a group of diffeomorphisms? In this talk, we will focus on diffeomorphisms of the closed interval in different regularities. In particular, we will present some natural obstructions to distortion (such as the presence of hyperbolic fixed points in $C^1$ regularity and the positivity of the so-called asymptotic variation in $C^2$ regularity (and higher)), and we will wonder whether they are the only ones.