## Conjugacy classes of diffeomorphisms of the interval in $\mathcal {C}^{1}$-regularity

### Volume 237 / 2017

#### Abstract

We consider the conjugacy classes of diffeomorphisms of the interval, endowed with the $\mathcal{C}^1$-topology. Given two diffeomorphisms $f,g$ of $[0;1]$ without hyperbolic fixed points, we give a complete answer to the following two questions:

$\bullet$ under what conditions does there exist a sequence of smooth conjugates $h_n f h_n^{-1}$ of $f$ tending to $g$ in the $\mathcal{C}^1$-topology?

$\bullet$ under what conditions does there exist a continuous path of $\mathcal{C}^1$-diffeomorphisms $h_t$ such that $h_t f h_t^{-1}$ tends to $g$ in the $\mathcal{C}^1$-topology?

We also present some consequences of these results to the study of $\mathcal{C}^1$-centralizers for $\mathcal{C}^1$-contractions of $[0;\infty)$; for instance, we exhibit a $\mathcal{C}^1$-contraction whose centralizer is uncountable and abelian, but is not a flow.