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Orbits of the Bernoulli measure in single-transition asynchronous cellular automata
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AP, Automata 2011 - 17th International Workshop on Cellular Automata and Discrete Complex Systems, DMTCS Proceedings vol. AP, Automata 2011 - 17th International Workshop on Cellular Automata and Discrete Complex Systems (2011).

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We study iterations of the Bernoulli measure under nearest-neighbour asynchronous binary cellular automata (CA) with a single transition. For these CA, we show that a coarse-level description of the orbit of the Bernoulli measure can be obtained, that is, one can explicitly compute measures of short cylinder sets after arbitrary number of iterations of the CA. In particular, we give expressions for probabilities of ones for all three minimal single-transition rules, as well as expressions for probabilities of blocks of length 3 for some of them. These expressions can be interpreted as "response curves'', that is, curves describing the dependence of the final density of ones on the initial density of ones. 
@article{DMTCS_2011_special_261_a4,
     author = {Fuk\'s, Henryk and Skelton, Andrew},
     title = {Orbits of the {Bernoulli} measure in single-transition asynchronous cellular automata},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AP, Automata 2011 - 17th International Workshop on Cellular Automata and Discrete Complex Systems},
     year = {2011},
     doi = {10.46298/dmtcs.2972},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2972/}
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Fukś, Henryk; Skelton, Andrew. Orbits of the Bernoulli measure in single-transition asynchronous cellular automata. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AP, Automata 2011 - 17th International Workshop on Cellular Automata and Discrete Complex Systems, DMTCS Proceedings vol. AP, Automata 2011 - 17th International Workshop on Cellular Automata and Discrete Complex Systems (2011). doi : 10.46298/dmtcs.2972. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2972/

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