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 ${\mathcal{A}\mathcal{C}}_{+}({\mathbb{S}}^1)$ be the group of absolutely continuous preserving orientation bijection with inverse absolutely continuous. We denote by $\mathcal{A}\mathcal{C}\mathcal{I}$ the group generated by $IET({\mathbb{S}}^1)$ and ${\mathcal{A}\mathcal{C}}_{+}({\mathbb{S}}^1)$. Given a suitable distance on $\mathcal{A}\mathcal{C}\mathcal{I}$, we classify all continuous homomorphisms $\rho :\mathbb{R}\to \mathcal{A}\mathcal{C}\mathcal{I}$. More precisely, $\rho$ is conjugated to a continuous homomorphism $\hat{\rho }:\mathbb{R}\to {\mathcal{A}\mathcal{C}}_{+}({\uplus }_i({\mathbb{S}}^1{)}_i)$.