(Arc-)connectedness for the space of $\mathbb{Z}^d$-actions by $C^2$ diffeomorphisms on 1-dimensional manifolds

TL;DR

This paper proves that the space of $Z^d$ actions by $C^2$ diffeomorphisms on 1-dimensional manifolds is connected, and any two such actions can be connected by a path of $C^{1+ac}$ actions, advancing understanding of regularity in dynamical systems.

Contribution

It establishes the connectedness of the space of $Z^d$ actions by $C^2$ diffeomorphisms and introduces arc-connectedness results for $C^{1+ac}$ actions, extending previous regularity bounds.

Findings

The space of $Z^d$ actions by $C^2$ diffeomorphisms on the interval is connected.

Any two $Z^d$ actions by $C^2$ diffeomorphisms are connected by a $C^{1+ac}$ path.

The results apply to all $Z^d$ actions by $C^{1+ac}$ diffeomorphisms without hyperbolic periodic points.

Abstract

We deal with the general problem of connectedness for the space of $\mathbb{Z}^d$ actions by (orientation-preserving) diffeomorphisms of a compact 1-manifold. We prove two results. First, the space of $\mathbb{Z}^d$ actions by $C^2$ diffeomorphisms of the interval is connected. Second, any two $\mathbb{Z}^d$ actions by $C^2$ diffeomorphisms of a compact 1-manifold are connected by a continuous path of $C^{1+\mathrm{ac}}$ actions (where $C^{1+ac}$ stands for diffeomorphisms with absolutely continuous derivative). The latter is the first result of arc-connectedness in regularity larger than $C^1$ in this setting. Actually, our proof applies to all $\mathbb{Z}^d$ actions by $C^{1+\mathrm{ac}}$ diffeomorphisms without elements with hyperbolic periodic points; the only obstruction to extend it to the general $C^{1+\mathrm{ac}}$ framework comes from the failure of the Sternberg-Yoccoz linearization theorem in class $C^{1+\mathrm{ac}}$.