Conjugacy classes of diffeomorphisms of the interval in C-1-regularity

We consider the conjugacy classes of diffeomorphisms of the interval, endowed with the C-1-topology. Given two diffeomorphisms f,g of [0; 1] without hyperbolic fixed points, we give a complete answer to the following two questions: under what conditions does there exist a sequence of smooth conjugates h(n)fh(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)fh(t)(-1) tends to g in the C-1-topology? We also present some consequences of these results to the study of C-1-centralizers for C-1-contractions of [0; infinity); for instance, we exhibit a C-1-contraction whose centralizer is uncountable and abelian, but is not a flow.