| This dissertation is a study of the relationship between minimal dynamical systems on the product of the Cantor set (X) and torus ( T2 ) and their corresponding crossed product C*-algebras.;For the case when the cocyles are rotations, we studied the structure of the crossed product C*-algebra A by looking at a large subalgebra Ax. It is proved that, as long as the cocyles are rotations, the tracial rank of the crossed product C*-algebra is always no more than one, which then indicates that it falls into the category of classifiable C*-algebras. In order to determine whether the corresponding crossed product C*-algebras of two such minimal dynamical systems are isomorphic or not, we just need to look at the Elliott invariants of these C*-algebras.;If a certain rigidity condition is satisfied, it is shown that the crossed product C*-algebra has tracial rank zero. Under this assumption, it is proved that for two such dynamical systems, if A and B are the corresponding crossed product C*-algebras, and we have an isomorphism between Ki( A) and Ki(B) which maps Ki(C(X x T2 )) to Ki(C(X x T2 )), then these two dynamical systems are approximately K-conjugate. The proof also indicates that C*-strongly flip conjugacy implies approximate K-conjugacy in this case.;We also studied the case when the cocyles are Furstenberg transformations, and some results on weakly approximate conjugacy and the K-theory of corresponding crossed product C*-algebras are obtained. |
