Distortion in One-Head Machines and Cellular Automata

Guillon, P.; Salo, Ville

doi:10.1007/978-3-319-58631-1_10

Cited by 13 publications

(17 citation statements)

References 17 publications

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“…In [15], a notion of range-distortion is defined. This notion is implied by distortion, and occurs in automorphism groups of all uncountable sofic shifts [27]. Since our group-embeddings are by simulation, it follows that PAut(A) also contains range-distorted elements.…”

Section: Finitely Subgenerated Cellular Automata Groupsmentioning

confidence: 96%

Groups and Monoids of Cellular Automata

Salo

2015

Cellular Automata and Discrete Complex Systems

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We prove that the group of reversible cellular automata (RCA), on any alphabet A, contains a perfect subgroup generated by six involutions which contains an isomorphic copy of every finitely-generated group of RCA on any alphabet B. This result follows from a case study of groups of RCA generated by symbol permutations and partial shifts with respect to a fixed Cartesian product decomposition of the alphabet. For prime alphabets, we show that this group is virtually cyclic, and that for composite alphabets it is non-amenable. For alphabet size four, it is a linear group. For non-prime non-four alphabets, it contains copies of all finitelygenerated groups of RCA. We also obtain that RCA of biradius one on all large enough alphabets generate copies of all finitely-generated groups of RCA. We ask a long list of questions.

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Section: Finitely Subgenerated Cellular Automata Groupsmentioning

confidence: 96%

Groups and Monoids of Cellular Automata

Salo

2015

Cellular Automata and Discrete Complex Systems

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“…In [14], a notion of range-distortion is defined. This notion is implied by distortion, and occurs in automorphism groups of all uncountable sofic shifts [25]. Since our group embeddings are by simulation, it is not hard to show that PAut(A) also contains range-distorted elements.…”

Section: The Abstract Statementmentioning

confidence: 96%

Universal groups of cellular automata

Salo

2022

Colloq. Math.

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We prove that the group of reversible cellular automata (RCA), on any alphabet A, contains a subgroup generated by three involutions which contains an isomorphic copy of every finitely generated group of RCA on any alphabet B. This result follows from a case study of groups of RCA generated by symbol permutations and partial shifts (equivalently, partitioned cellular automata) with respect to a fixed Cartesian product decomposition of the alphabet. For prime alphabets, we show that this group is virtually cyclic, and that for composite alphabets it is non-amenable. For alphabet size four, it is a linear group. For non-prime non-four alphabets, it contains copies of all finitely generated groups of RCA. We also prove this property for the group generated by RCA of biradius one on any full shift with large enough alphabet, and also for some perfect finitely generated groups of RCA.

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“…In the case of a single CA, we have seen that r − (t) ≥ r − t and r + (t) ≤ r + t. It is known that these sequences will be asymptotically linear, the slopes being called the Lyapunov exponents (see [13] for a discussion on possible growths for these sequences).…”

Section: For Everymentioning

confidence: 98%

The generic limit set of cellular automata

Djenaoui,

Guillon

2018

Preprint

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In topological dynamics, the generic limit set is the smallest closed subset which has a comeager realm of attraction. We study some of its topological properties, and the links with equicontinuity and sensitivity. We emphasize the case of cellular automata, for which the generic limit set is included in all subshift attractors, and discuss directional dynamics, as well as the link with measure-theoretical similar notions.

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Distortion in One-Head Machines and Cellular Automata

Cited by 13 publications

References 17 publications

Groups and Monoids of Cellular Automata

Groups and Monoids of Cellular Automata

Universal groups of cellular automata

The generic limit set of cellular automata

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