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This paper studies the relationship between minimal dynamical systems on theproduct of the Cantor set ($X$) and torus ($\T^2$) and their correspondingcrossed product $C^*$-algebras.  For the case when the cocycles are rotations, we studied the structure of thecrossed product $C^*$-algebra $A$ by looking at a large subalgebra $A_x$. It isproved that, as long as the cocycles are rotations, the tracial rank of thecrossed product $C^*$-algebra is always no more than one, which then indicatesthat it falls into the category of classifiable $C^*$-algebras.  If a certain rigidity condition is satisfied, it is shown that the crossedproduct $C^*$-algebra has tracial rank zero. Under this assumption, it isproved that for two such dynamical systems, if $A$ and $B$ are thecorresponding crossed product $C^*$-algebras, and we have an isomorphismbetween $K_i(A)$ and $K_i(B)$ which maps $K_i(C(X \times \T^2))$ to $K_i(C(X\times \T^2))$, then these two dynamical systems are approximately$K$-conjugate. The proof also indicates that $C^*$-strongly flip conjugacyimplies approximate $K$-conjugacy in this case.

Crossed Product $C^*$-algebras of Minimal Dynamical Systems on the Product of the Cantor Set and the Torus