Distortion elements in $Dif{f}^{\infty }(R/Z)$

Abstract

We consider the group of smooth diffeomorphisms of the circle. We show that any recurrent $f$ (in the sense that $\{f^n{\}}_{n\in Z}$ is not discrete) is in fact a distortion element (in the sense that its iterates can be written as short compositions involving finitely many smooth diffeomorphisms). Thus rotations are distortion elements.

5 pages