S. J. Kim and T. H. Kwon(1.2) have found that mixing in a single screw extrusion process can be enhanced by inserting spatially periodic barriers in the screw channel. They termed this new screw Chaos Screw, since the mixing due to this new screw is chaotic in nature. We exploited the connection between the kinematics of mixing and the theory of dynamical systems in a single-screw extrusion process with the Chaos Screw. From the study via Poin-care section we found that the periodic array of barrier is strongly related to the size of the islands. Continuous de-formation of a shell clearly shows the mechanism of chaotic mixing. namely repetition of stretching and folding, which induces the exponential growth of stretching in the chaotic flow. The local specific rate of stretching of a material line and the mixing efficiency can be evaluated in principle, but with some numerical difficulty. In contrast to a regular flow, the particle tracking is quite sensitive to the time step during the Runge-Kutta integration in the chaotic flow. Therefore the accuracy in determining the local rate of stretching and the mixing efficiency is not guaranteed with a realistic time step. In this regard, as a new measure of mixing, we propose σ_z which is related to the stretch rate of a relatively long line along the channel direction. The shorter the period of barrier-region is, the larger σ_z becomes. But according to the study via Poindcare section, when the period is too short, two large islands are found to exist. This fact shows that the large stretch rate of material does not always mean good mixing performance. In this respect, it is desirable to use the σ_z together with Poincare section in designing the mixing screw.