In this talk, we characterize the various kinds of mixing properties, in the sense of classical and complete prefix code (CPC for short) considerations, of the axial product of -shifts on a free semigroup G. Axial product space is an anisotropic Markov system which plays an essential role in the research of statistical physics. We reveal matrix criteria for examining these properties. Furthermore, the existence of the dense (CPC-)periodic orbits is also demonstrated. Combining this with the aforementioned results of the (CPC-) mixing properties exhibits whether a referred system is chaotic in the sense of Devaney.