Abstract
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. Noticeably, some CA are only transitive, but not mixing on their subsystems. 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. 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.
Highlights
For one-dimensional cellular automata (CA), this paper proves that the shift transitivity guarantees the CA transitivity and the CA with transitive non-trivial Bernoulli subshift of finite type have dense periodic points
It is proved that, for any 1-D CA restricted on its Bernoulli-shift subsystem, the shift transitivity implies the CA transitivity, and transitive nontrivial Bernoulli subshift of finite type (BSFT) has dense periodic points
1) Theorem 1 gives a convenient method to check if a CA f is transitive on a BSFT, since is transitive on SFT if and only if the transition matrix corresponding to the SFT is irreducible [23,24]
Summary
Cellular 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 [1]. In the late 1960s, Conway proposed his now-famous Game of Life, which shows the great potential of CA in simulating complex systems [2]. In the 1980s, Wolfram focused on the analysis of dynamical systems and studied CA in detail [3,4]. In 2002, he introduced the monumental work A New Kind of Science [5]. There are 40 topologically-distinct period- k rules k 1, 2,3, 6 , 30 topologically-distinct Bernoulli shift rules, 10 complex Bernoulli shift rules, and 8 hyper Bernoulli shift rules. Each ECA can be expressed by a 3-bit Boolean function and coded by an integer N , which is the decimal notation of the output binary sequence of the Boolean function [5,7,18]
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