provided a rigorous nonlinear dynamical approach to Wolfram’s empirical observations. In fact, mathematical theory of CA was firstly developed by Hedlund about two decades after Neumann’s seminal work. In 2002, he introduced the monumental work A New Kind of Science. In the 1980s, Wolfram focused on the analysis of dynamical systems and studied CA in detail. In the late 1960s, Conway proposed his now-famous Game of Life, which shows the great potential of CA in simulating complex systems. It is concluded that, for one-dimensional CA, the transitivity implies chaos in the sense of Devaney on the non-trivial Bernoulli subshift of finite types.Ĭellular automata (CA), formally introduced by von Neumann in the late 1940s and early 1950s, are a class of spatially and temporally discrete deterministic systems, characterized by local interactions and an inherently parallel form of evolution. Yet, for one-dimensional CA, this paper proves that not only the shift transitivity guarantees the CA transitivity but also the CA with transitive non-trivial Bernoulli subshift of finite type have dense periodic points. Noticeably, some CA are only transitive, but not mixing on their subsystems. Recent progress in symbolic dynamics of cellular automata (CA) shows that many CA exhibit rich and complicated Bernoulli-shift properties, such as positive topological entropy, topological transitivity and even mixing. Keywords: Bernoulli Subshift of Finite Type Cellular Automata Devaney Chaos Symbolic Dynamics Topological Transitivity 1School of Science, Hangzhou Dianzi University, Hangzhou, ChinaĢDepartment of Electronic Engineering, City University of Hong Kong, Hong Kong, ChinaģCollege of Pharmaceutical Sciences, Zhejiang Chinese Medical University, Hangzhou, ChinaĮmail: Novemrevised Decemaccepted January 14, 2013
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