Centralizers of C¹-contractions of the half line

الأصل · EN

A subgroup G⊂ Diff¹+([0,1]) is C¹-close to the identity if there is a sequence hₙ∈ Diff¹+([0,1]) such that the conjugates hₙ g hₙ⁻¹ tend to the identity for the C¹-topology, for every g∈ G. This is equivalent to the fact that G can be embedded in the C¹-centralizer of a C¹-contraction of [0,+∞) (see [Fa] and Theorem 1.1). We first describe the topological dynamics of groups C¹-close to the identity. Then, we show that the class of groups C¹-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⊂ Diff¹+([0,1]) whose topological dynamics implies that they are C¹-close to the identity. This allows us to show that the free group F₂ admits faithfull actions which are C¹-close to the identity. In particular, the C¹-centralizer of a C¹-contraction may contain free groups.