Conjugacy classes of diffeomorphisms of the interval in C1-regularity

In this paper we consider the conjugacy classes of diffeomorphisms of the interval, endowed with the $C^1$-topology. We present several results in the spirit of the one below : Given two diffeomorphisms $f,g$ of the interval $[0;1]$ without hyperbolic fixed point, we give a complete answer to the two following questions: - under what conditions does there exist a sequence of smooth conjugates $h_n f h_n^{-1}$ of $f$ tending to $g$ in the $C^1$-topology ? - under what conditions does there exist a continuous path of $C^1$-diffeomorphisms $h_t$ such that $h_t f h_t^{-1}$ tends to $g$ in the $C^1$ topology ? We present also some consequences of these results as regards the study of the $C^1$-centralizers of $C^1$-contractions of $[0;+\infty[$ ; for instance we exhibit a $C^1$-contraction whose centralizer is uncountable and abelian, but is not a flow.