Dynamical Cubes and a Criteria for Systems Having Product Extensions

For minimal Z2-topological dynamical systems, we introduce a cube structure and a variation of the regionally proximal relation for Z2 actions, which allow us to characterize product systems and their factors. We also introduce the concept of topological magic systems, which is the topological counterpart of measure theoretic magic systems introduced by Host in his study of multiple averages for commuting transformations. Roughly speaking, magic systems have a less intricate dynamic and we show that every minimal Z2 dynamical system has a magic extension. We give various applications of these structures, including the construction of some special factors in topological dynamics of Z2 actions, and a computation of the automorphism group of the minimal Robinson