The Space of C1+Ac Actions of Zd on a One-Dimensional Manifold Is Path-Connected

We show path-connectedness for the space of Zd actions by C1 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 Zd actions by C2 diffeomorphisms on the interval, as well as an analogous result in the real-analytic setting. © 2025 Elsevier Inc.