Continuous homomorphisms on piecewise absolutely continuous maps of $\mathbb{S}^{1}$

Abstract

Let $IET(\mathbb{S}^{1})$ be the group of interval exchange transformation of $\mathbb{S}^{1}$ and $\mathcal{AC}_{+}(\mathbb{S}^{1})$ be the group of absolutely continuous preserving orientation bijection with inverse absolutely continuous. We denote by $\mathcal{ACI}$ the group generated by $IET(\mathbb{S}^{1})$ and $\mathcal{AC}_{+}(\mathbb{S}^{1})$. Given a suitable distance on $\mathcal{ACI}$, we classify all continuous homomorphisms $\rho:\mathbb{R} \to \mathcal{ACI}$. More precisely, $\rho$ is conjugated to a continuous homomorphism $\hat{\rho}:\mathbb{R} \to \mathcal{AC}_{+}(\uplus_{i}(\mathbb{S}^{1})_{i})$.