E. Militon explained to the Groupe de travail de théorie ergodique in Orsay the following result.
Let be a group. Given and , the length is the minimum length of a product of elements of equal to (possibly ).
Definition. is a distortion element if there exists a finite subset , such that .
Let be the group of -smooth diffeomorphisms of a compact manifold which, moreover, are isotopic to the identity. Let $d$ be a metric on which is compatible with the -topology.
Theorem (Militon). If is recurrent, i.e., , then is a distortion element.
Avila proved a similar result in the case and Militon’s proof follows Avila’s.
Lemma 1. There exist two numerical sequences such that for any sequence of diffeomorphisms satisfying there exists a finite subset such that for all .
This Lemma is easily seen to imply the theorem. It is deduced from the next lemma using a non-trivial result on the decomposition of a diffeomorphism close enough to the identity into a composition of a bounded number of commutators of diffeomorphisms with small supports and themselves close to the identity.
denotes the set of smooth diffeomorphisms of which are compact supported and are isotopic to the identity through a path of diffeomorphisms with supports all included in a fixed compact set of $\mathbb R^d$.
Lemma 2. There exist two numerical sequences such that for any pair of sequences of diffeomorphisms satisfying , (and similar conditions on ), there exists a finite subset such that for all .