Topological Conjugacies Between Cellular Automata
We study cellular automata as discrete dynamical systems and in particular investigate under which conditions two cellular automata are topologically conjugate. Based on work of McKinsey, Tarski, Pierce and Head we introduce derivative algebras to study the topological structure of sofic shifts in...
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Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden
2017
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ndltd-DRESDEN-oai-qucosa.de-bsz-14-qucosa-2318232017-12-20T03:32:05Z Topological Conjugacies Between Cellular Automata Epperlein, Jeremias Zelluläre Automaten Topologische Konjugation Ableitungsalgebra Dynamische Systeme Entropie Symbolische Dynamik cellular automata topogical conjugacy derivative algebra dynamical systems entropy subshifts symbolic dynamics ddc:510 rvk:SK 950 rvk:ST 136 Symbolische Dynamik Dynamisches System Zellularer Automat We study cellular automata as discrete dynamical systems and in particular investigate under which conditions two cellular automata are topologically conjugate. Based on work of McKinsey, Tarski, Pierce and Head we introduce derivative algebras to study the topological structure of sofic shifts in dimension one. This allows us to classify periodic cellular automata on sofic shifts up to topological conjugacy based on the structure of their periodic points. We also get new conjugacy invariants in the general case. Based on a construction by Hanf and Halmos, we construct a pair of non-homeomorphic subshifts whose disjoint sums with themselves are homeomorphic. From this we can construct two cellular automata on homeomorphic state spaces for which all points have minimal period two, which are, however, not topologically conjugate. We apply our methods to classify the 256 elementary cellular automata with radius one over the binary alphabet up to topological conjugacy. By means of linear algebra over the field with two elements and identities between Fibonacci-polynomials we show that every conjugacy between rule 90 and rule 150 cannot have only a finite number of local rules. Finally, we look at the sequences of finite dynamical systems obtained by restricting cellular automata to spatially periodic points. If these sequences are termwise conjugate, we call the cellular automata conjugate on all tori. We then study the invariants under this notion of isomorphism. By means of an appropriately defined entropy, we can show that surjectivity is such an invariant. Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden Technische Universität Dresden, Fakultät Mathematik und Naturwissenschaften Prof. Dr. Stefan Siegmund Prof. Dr. Stefan Siegmund Prof. Dr. Jarkko Kari 2017-12-19 doc-type:doctoralThesis application/pdf http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-231823 urn:nbn:de:bsz:14-qucosa-231823 http://www.qucosa.de/fileadmin/data/qucosa/documents/23182/dissertation-pdfa.pdf eng |
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NDLTD |
language |
English |
format |
Doctoral Thesis |
sources |
NDLTD |
topic |
Zelluläre Automaten Topologische Konjugation Ableitungsalgebra Dynamische Systeme Entropie Symbolische Dynamik cellular automata topogical conjugacy derivative algebra dynamical systems entropy subshifts symbolic dynamics ddc:510 rvk:SK 950 rvk:ST 136 Symbolische Dynamik Dynamisches System Zellularer Automat |
spellingShingle |
Zelluläre Automaten Topologische Konjugation Ableitungsalgebra Dynamische Systeme Entropie Symbolische Dynamik cellular automata topogical conjugacy derivative algebra dynamical systems entropy subshifts symbolic dynamics ddc:510 rvk:SK 950 rvk:ST 136 Symbolische Dynamik Dynamisches System Zellularer Automat Epperlein, Jeremias Topological Conjugacies Between Cellular Automata |
description |
We study cellular automata as discrete dynamical systems and in particular investigate under which conditions two cellular automata are topologically conjugate.
Based on work of McKinsey, Tarski, Pierce and Head we introduce derivative algebras to study the topological structure of sofic shifts in dimension one. This allows us to classify periodic cellular automata on sofic shifts up to topological conjugacy based on the structure of their periodic points. We also get new conjugacy invariants in the general case. Based on a construction by Hanf and Halmos, we construct a pair of non-homeomorphic subshifts whose disjoint sums with themselves are homeomorphic. From this we can construct two cellular automata on homeomorphic state spaces for which all points have minimal period two, which are, however, not topologically conjugate. We apply our methods to classify the 256 elementary cellular automata with radius one over the binary alphabet up to topological conjugacy. By means of linear algebra over the field with two elements and identities between Fibonacci-polynomials we show that every conjugacy between rule 90 and rule 150 cannot have only a finite number of local rules. Finally, we look at the sequences of finite dynamical systems obtained by restricting cellular automata to spatially periodic points. If these sequences are termwise conjugate, we call the cellular automata conjugate on all tori. We then study the invariants under this notion of isomorphism. By means of an appropriately defined entropy, we can show that surjectivity is such an invariant. |
author2 |
Technische Universität Dresden, Fakultät Mathematik und Naturwissenschaften |
author_facet |
Technische Universität Dresden, Fakultät Mathematik und Naturwissenschaften Epperlein, Jeremias |
author |
Epperlein, Jeremias |
author_sort |
Epperlein, Jeremias |
title |
Topological Conjugacies Between Cellular Automata |
title_short |
Topological Conjugacies Between Cellular Automata |
title_full |
Topological Conjugacies Between Cellular Automata |
title_fullStr |
Topological Conjugacies Between Cellular Automata |
title_full_unstemmed |
Topological Conjugacies Between Cellular Automata |
title_sort |
topological conjugacies between cellular automata |
publisher |
Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden |
publishDate |
2017 |
url |
http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-231823 http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-231823 http://www.qucosa.de/fileadmin/data/qucosa/documents/23182/dissertation-pdfa.pdf |
work_keys_str_mv |
AT epperleinjeremias topologicalconjugaciesbetweencellularautomata |
_version_ |
1718564159167135744 |