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Centralizers of $C^1$-contractions of the half line

A subgroup $G\subset Diff^1_+([0,1])$ is $C^1$-close to the identity if there is a sequence $h_n\in Diff^1_+([0,1])$ such that the conjugates $h_n g h_n^{-1}$ tend to the identity for the $C^1$-topology, for every $g\in G$. This is equivalent to the fact that $G$ can be embedded in the $C^1$-centralizer of a $C^1$-contraction of $[0,+\infty)$ (see [Fa] 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 $G\subset Diff^1_+([0,1])$ whose topological dynamics implies that they are $C^1$-close to the identity. This allows us to show that the free group ${\mathbb F}_2$ admits faithfull actions which are $C^1$-close to the identity. In particular, the $C^1$-centralizer of a $C^1$-contraction may contain free groups.