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## E. Militon: Distortion elements in groups of smooth diffeomorphisms

E. Militon explained to the Groupe de travail de théorie ergodique in Orsay the following result.

Let $G$ be a group. Given $S\subset G$ and $g\in G$, the length $\ell_S(g)$ is the minimum length of a product of elements of $S\cup S^{-1}$ equal to $g$ (possibly $\infty$).

Definition. $g\in G$ is a distortion element if there exists a finite subset $S\subset G$, such that $\lim_{n\to\infty} \ell_S(g^n)/n = 0$.

Let $G=Diff^\infty_0(M)$ be the group of $C^\infty$-smooth diffeomorphisms of a compact manifold $M$ which, moreover, are isotopic to the identity. Let $d$ be a metric on $G$ which is compatible with the $C^\infty$-topology.

Theorem (Militon). If $g\in G$ is recurrent, i.e., $\liminf_{n\to\infty} d(g^n,Id)=0$, then $g$ is a distortion element.

Avila proved a similar result in the case $M=\mathbb S^1$ and Militon’s proof follows Avila’s.

Lemma 1. There exist two numerical sequences $\epsilon_n,k_n$ such that for any sequence of diffeomorphisms $h_n\in Diff^\infty_0(M)$ satisfying $d(h_n,Id)<\epsilon_n$ there exists a finite subset $S\subset G$ such that $\ell_S(h_n)\leq k_n$ for all $n\geq1$.

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.

$Diff^\infty_0(\mathbb R^d)$ denotes the set of smooth diffeomorphisms of $\mathbb R^d$ 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 $\epsilon_n,k_n$ such that for any pair of sequences of diffeomorphisms $f_n,g_n\in Diff^\infty_0(\mathbb R^d)$ satisfying $d(f_n,Id)<\epsilon_n$, $supp(f_n)\subset B(0,1)$ (and similar conditions on $g_n$), there exists a finite subset $S\subset G$ such that $\ell_S([f_n,g_n])\leq k_n$ for all $n\geq1$.

## T. Fisher: Diffeomorphisms with Trivial Centralizers

A major part of dynamical system theory studies the iteration of diffeomorphisms under various assumptions, geometric or analytic or otherwise. It is also interesting to study them collectively. In particular, $Diff^r(M)$, the set of $C^r$ diffeomorphisms of some manifold $M$ and some number $r\geq1$, is a group. It is for instance of interest to know when it is simple.

Todd Fisher gave a talk in Orsay about the (non)triviality of the centralizer for many elements. Recall that the centralizer of an element $g$ of a group $G$ is $Z(g):=\{h\in G:hg=gh\}$, the set of all elements that commute with $g$. It is also the set of all elements $h$ that conjugate $g$ with itself, i.e., it can be interpreted as the set of symmetries of $g$.

$Z(g)$ may be very large, e.g., equal to the whole of $G$ if this group is Abelian. It is said to be trivial if it is reduced to $\{g^n:n\in\mathbb Z\}$ (which is always part of it). Note that such a diffeomorphism cannot be embedded in a smooth flow.

One expects that the set $\mathcal T^r(M)$ of $C^r$-diffeomorphisms of $M$ with trivial centralizer is very big (residual or even open and dense). To show this for all compact manifolds and all $1\leq r\leq\infty$) was one of Smale’s questions for the the twenty-first century.

Results are known in dimension one or $C^1$-smoothness under hyperbolicity assumptions:

• $\mathcal T^r(\mathbb S^1)$ is open and dense for all $2\leq r\leq\infty$ (Kopell);
• a generic $C^1$ diffeomorphism of an arbitrary manifold has trivial $C^1$-centralizer (Bonatti, Crovisier, Wilkinson 2008).

It has turned out that the above is really only generic and fails to hold on an open and dense set according to Bonatti, Crovisier, Vago and Wilkinson. On the circle, this holds only on a set with empty interior. This last result extends to tori in dimensions up to 4 according to a work in progress of Bakker and Fisher.

Hyperbolicity also allows some results:

• a generic $C^1$-axiom A diffeomorphisms with no cycles has trivial centralizer (Tugawa 1978);
• there is an open and dense set of surface $C^\infty$ axiom-A diffeomorphisms with no cycles having a trivial $C^1$-centralizer (Palis-Yoccoz 1989). This even holds for arbitrary dimensions if one assumes the existence of a sink or a source.

On surfaces, Fisher has extended the last Palis-Yoccoz result to intermediate smoothness, i.e., $C^r$, $2\leq r\leq \infty$ and axiom A with no-cycles, i.e., $\Omega$-stable, diffeomorphisms instead of the above ones with strong transversality, i.e., the structurally stable ones.

The proof of Palis and Yoccoz (and of its later generalizations) relies on a local theorem showing that on each basin, two commuting maps must be iterates one of the other and a global theorem that connects the basins and identify the iterates. The last is the delicate part of the proof.

## Dynamique de pseudo-groupes discrets agissant de façon expansive et holomorphe

Bertrand DEROIN a présenté une extension au cadre holomorphe d’un théorème de Duminy (dont une preuve figure dans le livre d’A. Navas).

Un pseudo-groupe agissant sur un espace topologique $X$ est une collection d’homéomorphisme entre ouverts de $X$ stable par composition, inversion et restriction.

Théorème (B. Deroin). Considérons un pseudo-groupe agissant de façon holomorphe, discrète et expansive au voisinage d’un certain fermé invariant d’une certaine surface de Riemann. Alors:

• ou bien on a un pseudo-groupe d’applications à allure polynomiale généralisée;
• ou bien on a (à conjugaison analytique près) un groupe Kleinien.