Surjunctivity and Reversibility of Cellular Automata over Concrete Categories

Ceccherini‐Silberstein, Tullio; Coornaert, Michel

doi:10.1007/978-88-470-2853-1_6

Cited by 7 publications

(2 citation statements)

References 31 publications

(31 reference statements)

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“…CAs with alphabet E = R are applicable to modeling the heat equation [18]. Theorems about surjectivity of CAs have been extended to CAs whose alphabets are (possibly infinite) objects in some concrete category and then guarantee that some CAs with infinite alphabets have a Garden of Eden configuration (a configuration that does not have a predecessor) [7]. Recently, complex PCAs with infinite and continuous alphabets have been proposed in [20] in order to model urban dynamics.…”

Section: Pcas With General Alphabetsmentioning

confidence: 99%

Probabilistic cellular automata with general alphabets possessing a Markov chain as an invariant distribution

Casse

2016

Adv. Appl. Probab.

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This paper is devoted to probabilistic cellular automata (PCA) on N, Z or Z/nZ, depending of two neighbors, with a general alphabet E (finite or infinite, discrete or not).We study the following question: under which conditions does a PCA possess a Markov chain as invariant distribution? Previous results in the literature give some conditions on the transition matrix (for positive rate PCA) when the alphabet E is finite. Here we obtain conditions on the transition kernel of PCA with a general alphabet E. In particular, we show that the existence of an invariant Markov chain is equivalent to the existence of a solution to a cubic integral equation.One of the difficulties to pass from a finite alphabet to a general alphabet comes from some problems of measurability, and a large part of this work is devoted to clarify these issues.

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Section: Pcas With General Alphabetsmentioning

confidence: 99%

Probabilistic cellular automata with general alphabets possessing a Markov chain as an invariant distribution

Casse

2016

Adv. Appl. Probab.

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“…Thus, the shift system (S Γ , Γ) is never surjunctive when the alphabet S is compressible. If M is a closed topological manifold and the group Γ is residually finite, it was observed in [3,Corollary 7.8] that the Γ-shift (M Γ , Γ) is surjunctive. On the other hand, it follows from a deep theorem of Gromov [6] and Weiss [13] that, if Γ is a sofic group and S is a finite discrete space, then the Γshift (S Γ , Γ) is surjunctive.…”

Section: Introductionmentioning

confidence: 99%