Guan, Junbiao; Shen, Shaowei; Tang, Changbing; Chen, Fangyue Extending Chua’s global equivalence theorem on Wolfram’s new kind of science. (English) Zbl 1143.37303 Int. J. Bifurcation Chaos Appl. Sci. Eng. 17, No. 12, 4245-4259 (2007). 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. Cited in 9 Documents MSC: 37B15 Dynamical aspects of cellular automata Keywords:Cellular Automata; equivalence class; symbolic dynamics; topological conjugacy Citations:Zbl 1091.37500; Zbl 1140.37001 PDFBibTeX XMLCite \textit{J. Guan} et al., Int. J. Bifurcation Chaos Appl. Sci. Eng. 17, No. 12, 4245--4259 (2007; Zbl 1143.37303) Full Text: DOI References: [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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.