Laboratoire de Mathématiques Jean Leray

Given a result of M. Herman, we provide a new elementary proof of the fact that the connected component of the group of compactly supported diffeomorphisms is perfect and hence simple. Moreover, we show that every diffeomorphism g, which is sufficiently close to the identity, can be represented as a product of four commutators, g = [h_1 , k_1 ]· · ·[h4 , k4 ], where the factors h_i and k_i can be chosen to depend smoothly on g. Next the obtained results will be compared with classical and recent theorems on the structure of diffeomorphism groups.