Approximate $K$-conjugacies and $C^*$-approximate conjugacies of minimal dynamical systems

TL;DR

This paper extends the concepts of approximate $K$-conjugacies and $C^*$-approximate conjugacies to general minimal dynamical systems, establishing conditions under which these notions coincide for certain skew product systems.

Contribution

It generalizes previous notions to broader minimal systems and shows the equivalence of different conjugacy concepts for specific skew product classes.

Findings

Approximate $K$-conjugacy and $C^*$-strongly approximate conjugacy coincide for certain skew products.

Constructs a class of minimal skew products with specific properties.

Answers a question of H. Lin regarding conjugacy notions in minimal systems.

Abstract

In this article, we extend H. Matui and H. Lin's notions of approximate $K$-conjugacies and $C^*$-strongly approximate conjugacies to general minimal dynamical systems. In particular, upon modifying a result of the existence of minimal skew products, we answer a question of H. Lin and show that, associated with any Cantor minimal system $(K,\tilde{\alpha})$, there is a class $R_0(\tilde{\alpha})$ of minimal skew products on $K\times\Omega$, such that for any two rigid homeomorphisms $\alpha\in R_0(\tilde{\alpha})$ and $\beta\in R_0(\tilde{\beta})$, the notions of approximate $K$-conjugacy and $C^*$-strongly approximate conjugacy coincide, which are also equivalent to a $K$-version of Tomiyama's commutative diagram, where $\Omega$ is an (infinite) connected finite CW-complex with torsion free $K$-groups and the so-called Lipschitz-minimal-property (LMP).