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

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).