# Centralizers of $C_{1}$-contractions of the half line

### Christian Bonatti

Université de Bourgogne, Dijon, France### Églantine Farinelli

Université de Bourgogne, Dijon, France

## Abstract

A subgroup $G⊂Diff_{+}([0,1])$ is *$C_{1}$-close to the identity* if there is a sequence $h_{n}∈Diff_{+}([0,1])$ such that the conjugates $h_{n}gh_{n}$ tend to the identity for the $C_{1}$-topology, for every $g∈G$. This is equivalent to the fact that $G$ can be *embedded in the $C_{1}$-centralizer* of a $C_{1}$-contraction of $[0,+∞)$ (see [6] and Theorem 1.1).

We first describe the topological dynamics of groups $C_{1}$-close to the identity. Then, we show that the class of groups $C_{1}$-close to the identity is invariant under some natural dynamical and algebraic extensions. As a consequence, we can describe a large class of groups $G⊂Diff_{+}([0,1])$ whose topological dynamics implies that they are $C_{1}$-close to the identity.

This allows us to show that the free group $F_{2}$ admits faithful actions which are $C_{1}$-close to the identity. In particular, the $C_{1}$-centralizer of a $C_{1}$-contraction may contain free groups.

## Cite this article

Christian Bonatti, Églantine Farinelli, Centralizers of $C_{1}$-contractions of the half line. Groups Geom. Dyn. 9 (2015), no. 3, pp. 831–889

DOI 10.4171/GGD/330