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.

Figures (7)

• Figure 1: The initial vector field $X_0$.
• Figure 2: One brick.
• Figure 3: The function $\delta$.
• Figure 4: Checking that $f^{n_k}= f_0^{n_k}$ from $J_k$ to $J_{k+1}$.
• Figure 5: Checking that $f^{\frac{1}{2}}= T_{-2^{-k^2-1}}\circ\varphi_k$ on $I_k^+$.

Theorems & Definitions (81)

• Remark 1.1
• Proposition 2.1
• proof
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Remark 2.5
• Remark 2.6
• proof
• Remark 3.2