The space of $C^{1+ac}$ actions of ${\mathbb{Z}}^d$ on a one-dimensional manifold is path-connected

Abstract

We show path-connectedness for the space of ${\mathbb{Z}}^d$ actions by $C^1$ diffeomorphisms with absolutely continuous derivative on both the closed interval and the circle. We also give a new and short proof of the connectedness of the space of ${\mathbb{Z}}^d$ actions by $C^2$ diffeomorphisms on the interval, as well as an analogous result in the real-analytic setting.

57 pages, 7 figures. This article improves and simplifies the results of arXiv:2103.06940, which will thus remain unpublished. This version contains a new Appendix in collaboration with Théo Virot concerning a general discussion on the topology of low-regular diffeomorphism groups and a technical lemma of analysis