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

TL;DR

This paper classifies all continuous homomorphisms from the real line into a group generated by interval exchange transformations and absolutely continuous maps on the circle, showing they are conjugate to homomorphisms into a simpler subgroup.

Contribution

It provides a complete classification of continuous homomorphisms from  into the group generated by interval exchange transformations and absolutely continuous maps on the circle.

Findings

All continuous homomorphisms are conjugate to those into  of a disjoint union of circles.

The classification relies on a suitable metric on the group .

The result extends understanding of the structure of homomorphisms in groups of circle transformations.

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 $\hat{\rho}:\mathbb{R} \to \mathcal{AC}_{+}(\uplus_{i}(\mathbb{S}^{1})_{i})$.