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Extending Chua’s global equivalence theorem on Wolfram’s new kind of science. (English) Zbl 1143.37303

Summary: We establish the relation between the extended (i.e. \(I = \infty\)) one-dimensional binary Cellular Automata (1D CA) and the bi-infinite symbolic sequences in symbolic dynamics. That is, the 256 local rules of 1D CA correspond to 256 local rule mappings in the symbolic space. By employing the two homeomorphisms \(T^†\) and \(\overline T\) from [L. O. Chua, V. I. Sbitnev and S. Yoon, Int. J. Bifurcation Chaos Appl. Sci. Eng. 3689–3820 (2004; Zbl 1091.37500)] for finite \(I\), we classify these 256 local rule mappings into the same 88 equivalence classes identified in [loc. cit.] and [ibid. 16, No. 5, 1097–1373 (2006; Zbl 1140.37001)]. Different mappings in the same equivalence class are mutually topologically conjugate.

MSC:

37B15 Dynamical aspects of cellular automata
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[1] DOI: 10.1142/S0218127402006333 · Zbl 1043.37009 · doi:10.1142/S0218127402006333
[2] DOI: 10.1142/S0218127403008041 · Zbl 1046.37004 · doi:10.1142/S0218127403008041
[3] DOI: 10.1142/S0218127404011764 · Zbl 1091.37500 · doi:10.1142/S0218127404011764
[4] DOI: 10.1142/S0218127405012995 · Zbl 1084.37011 · doi:10.1142/S0218127405012995
[5] DOI: 10.1142/S0218127405014775 · Zbl 1094.37500 · doi:10.1142/S0218127405014775
[6] Chua L. O., A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science 1 (2006) · Zbl 1095.37005
[7] DOI: 10.1142/S0218127406015544 · Zbl 1140.37001 · doi:10.1142/S0218127406015544
[8] DOI: 10.1007/978-3-642-58822-8 · doi:10.1007/978-3-642-58822-8
[9] Neumann J. V., Theory of Self-Reproducing Automata (1966)
[10] DOI: 10.1007/978-1-4757-4067-7 · doi:10.1007/978-1-4757-4067-7
[11] DOI: 10.1016/0167-2789(84)90245-8 · doi:10.1016/0167-2789(84)90245-8
[12] Wolfram S., A New Kind of Science (2002) · Zbl 1022.68084
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