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

TL;DR

This paper proves the path-connectedness of the space of certain smooth group actions on one-dimensional manifolds, extending known results to less smooth and real-analytic cases.

Contribution

It establishes path-connectedness for $C^{1+ac}$ actions of $bZ^d$ on intervals and circles, and provides new proofs for $C^2$ and real-analytic cases.

Findings

Path-connectedness of $C^{1+ac}$ actions on intervals and circles.

Simplified proof of connectedness for $C^2$ actions.

Extension of connectedness results to real-analytic actions.

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.