Groups, Geometry, and Dynamics

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Volume 9, Issue 3, 2015, pp. 831–889
DOI: 10.4171/GGD/330

Published online: 2015-10-29

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

Christian Bonatti[1] and Églantine Farinelli[2]

(1) Université de Bourgogne, Dijon, France
(2) Université de Bourgogne, Dijon, France

A subgroup $G\subset \operatorname{Diff}^1_+([0,1])$ is $C^1$-close to the identity if there is a sequence $h_n\in \operatorname{Diff}^1_+([0,1])$ 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 [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 $G\subset \operatorname{Diff}^1_+([0,1])$ 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 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.

Keywords: Actions on 1-manifolds, free groups, centralizer, translation number, $C^1$-diffeomorphisms

Bonatti Christian, Farinelli Églantine: Centralizers of $C^1$-contractions of the half line. Groups Geom. Dyn. 9 (2015), 831-889. doi: 10.4171/GGD/330