Chaotic properties of elementary cellular automata with majority memory

In this paper, a practical framework of symbolic vector space is applied to uncover the time-asymptotic evolutionary behaviors of cellular automata with majority memory. This work focuses on elementary cellular automata rules with majority memory (ECAMs) and Bernoulli-shift parameters σ=1,τ=2. The concepts of forward time−τ map and characteristic function are exploited to display the Bernoulli-shift features and modes. Particularly, it is rigorously verified that ECAMs rule 10 actually defines a Bernoulli-measure global attractor in the bi-infinite symbolic vector space. It is furthermore identified that ECAMs rule 10 possesses complicated symbolic dynamics; namely, it is endowed with temporal chaotic features as positive topological entropy and topologically mixing. Therefore, ECAMs rule 10 is chaotic on its global attractor according to definitions of both Li-York and Devaney. To this end, it should be underlined that the procedure proposed in this study is applied to other ECAMs rules with the same shifting mode, and the corresponding results are exhibited.