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Entropy 2014, 16(11), 5601-5617; doi:10.3390/e16115601

On One-Sided, D-Chaotic CA Without Fixed Points, Having Continuum of Periodic Points With Period 2 and Topological Entropy log(p) for Any Prime p

1
Institute of Computer Science, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland
2
Department of Computer Science and Econometrics, Silesian Technical University, Roosevelta 26-28, 41-800 Zabrze, Poland
*
Author to whom correspondence should be addressed.
Received: 18 May 2014 / Revised: 28 September 2014 / Accepted: 15 October 2014 / Published: 24 October 2014
(This article belongs to the Section Complexity)
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Abstract

A method is known by which any integer \(\, n\geq2\,\) in a metric Cantor space of right-infinite words \(\,\tilde{A}_{n}^{\,\mathbb N}\,\) gives a construction of a non-injective cellular automaton \(\,(\tilde{A}_{n}^{\,\mathbb N},\,\tilde{F}_{n}),\,\) which is chaotic in Devaney sense, has a radius \(\, r=1,\,\) continuum of fixed points and topological entropy \(\, log(n).\,\) As a generalization of this method we present for any integer \(\, n\geq2,\,\) a construction of a cellular automaton \(\,(A_{n}^{\,\mathbb{N}},\, F_{n}),\,\) which has the listed properties of \(\,(\tilde{A}_{n}^{\,\mathbb N},\,\tilde{F}_{n}),\,\) but has no fixed points and has continuum of periodic points with the period 2. The construction is based on properties of cellular automaton introduced here \(\,(B^{\,\mathbb N},\, F)\,\) with radius \(1\) defined for any prime number \(\, p.\,\) We prove that \(\,(B^{\,\mathbb N},\, F)\,\) is non-injective, chaotic in Devaney sense, has no fixed points, has continuum of periodic points with the period \(2\) and topological entropy \(\, log(p).\,\) View Full-Text
Keywords: one-sided cellular automata; D-chaotic; E-chaotic; fixed points; topological entropy one-sided cellular automata; D-chaotic; E-chaotic; fixed points; topological entropy
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Forys, W.; Matyja, J. On One-Sided, D-Chaotic CA Without Fixed Points, Having Continuum of Periodic Points With Period 2 and Topological Entropy log(p) for Any Prime p. Entropy 2014, 16, 5601-5617.

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