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

Églantine Farinelli¹

¹ Institut de Mathématiques de Bourgogne CNRS, URM 5584 Université de Bourgogne Dijon 21000, France

Fundamenta Mathematicae, Tome 237 (2017) no. 3, pp. 201-248.

Résumé

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.

Détail

DOI : 10.4064/fm594-8-2014

Keywords: consider conjugacy classes diffeomorphisms interval endowed mathcal topology given diffeomorphisms without hyperbolic fixed points complete answer following questions bullet under what conditions does there exist sequence smooth conjugates h tending mathcal topology bullet under what conditions does there exist continuous path mathcal diffeomorphisms h tends mathcal topology present consequences these results study mathcal centralizers mathcal contractions infty instance exhibit mathcal contraction whose centralizer uncountable abelian flow