Generation and genericity of the group of absolutely continuous homeomorphisms of the interval

TL;DR

This paper studies the group of order-preserving homeomorphisms of the interval that are absolutely continuous, showing it is topologically 2-generated and has a dense conjugacy class, revealing its rich algebraic and topological structure.

Contribution

It proves that the group is topologically 2-generated and has a dense conjugacy class, providing new insights into its algebraic and topological properties.

Findings

The group is topologically 2-generated.

There exists a dense G_delta conjugacy class in the group.

The elements of the conjugacy class are explicitly characterized.

Abstract

We examine the Polish group $H_{+}^{AC}$ of order-preserving self-homeomorphisms $f$ of the interval $\left[ 0,1 \right]$ for which both $f$ and $f^{-1}$ are absolutely continuous; in particular, we establish two results. First, we prove that $H_{+}^{AC}$ is topologically $2$-generated; in fact, it is generically $2$-generated, i.e., there is a dense $G_{\delta}$ set of pairs $\left( f,g \right) \in H_{+}^{AC} \times H_{+}^{AC}$ for which $\left\langle f,g \right\rangle$ is dense. Secondly, we prove that $H_{+}^{AC}$ admits a dense $G_{\delta}$ conjugacy class, and we explicitly characterize the elements thereof.