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
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 radi...
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| Format: | Article |
| Language: | English |
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MDPI AG
2014-10-01
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| Series: | Entropy |
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| Online Access: | http://www.mdpi.com/1099-4300/16/11/5601 |
| _version_ | 1798038829138968576 |
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| author | Wit Forys Janusz Matyja |
| author_facet | Wit Forys Janusz Matyja |
| author_sort | Wit Forys |
| collection | DOAJ |
| description | 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 \(\, log(p).\,\) |
| first_indexed | 2024-04-11T21:45:42Z |
| format | Article |
| id | doaj.art-2921b83825d147e0811f8f3840ad766f |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-04-11T21:45:42Z |
| publishDate | 2014-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-2921b83825d147e0811f8f3840ad766f2022-12-22T04:01:26ZengMDPI AGEntropy1099-43002014-10-0116115601561710.3390/e16115601e16115601On One-Sided, D-Chaotic CA Without Fixed Points, Having Continuum of Periodic Points With Period 2 and Topological Entropy log(p) for Any Prime pWit Forys0Janusz Matyja1Institute of Computer Science, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, PolandDepartment of Computer Science and Econometrics, Silesian Technical University, Roosevelta 26-28, 41-800 Zabrze, PolandA 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 \(\, log(p).\,\)http://www.mdpi.com/1099-4300/16/11/5601one-sided cellular automataD-chaoticE-chaoticfixed pointstopological entropy |
| spellingShingle | Wit Forys Janusz Matyja 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 one-sided cellular automata D-chaotic E-chaotic fixed points topological entropy |
| title | 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 |
| title_full | 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 |
| title_fullStr | 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 |
| title_full_unstemmed | 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 |
| title_short | 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 |
| title_sort | 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 |
| topic | one-sided cellular automata D-chaotic E-chaotic fixed points topological entropy |
| url | http://www.mdpi.com/1099-4300/16/11/5601 |
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