Approximate $K$-conjugacies and $C^{\ast }$-approximate conjugacies of minimal dynamical systems

Abstract

In this article, we extend H. Matui and H. Lin's notions of approximate $K$-conjugacies and $C^{\ast }$-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 $C^{\ast }$-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).

35 pages