Minimal dynamical systems on the product of the Cantor set and the circle II

Let $X$ be the Cantor set and $\phi$ be a minimal homeomorphism on $X\times\T$. We show that the crossed product $C^*$-algebra $C^*(X\times\T,\phi)$ is a simple $A\T$-algebra provided that the associated cocycle takes its values in rotations on $\T$. Given two minimal systems $(X\times\T,\phi)$ and $(Y\times\T,\psi)$ such that $\phi$ and $\psi$ arise from cocycles with values in isometric homeomorphisms on $\T$, we show that two systems are approximately $K$-conjugate when they have the same $K$-theoretical information.