Centralizers of C-1-contractions of the half line

A subgroup G subset of Diff(+)(1) ([0, 1]) is C-1-close to the identity if there is a sequence h(n) is an element of Diff(+)(1) ([0, 1]) such that the conjugates h(n)gh(n)(-1) tend to the identity for the C-1-topology, for every g is an element of G. This is equivalent to the fact that G can be embedded in the C-1-centralizer of a C-1-contraction of [0, +infinity)(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 of Diffi(+)(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 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.