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Licensed Unlicensed Requires Authentication Published online by De Gruyter January 31, 2024

The algebraic entropy of one-dimensional finitary linear cellular automata

  • Hasan Akın , Dikran Dikranjan , Anna Giordano Bruno ORCID logo EMAIL logo and Daniele Toller
From the journal Journal of Group Theory

Abstract

The aim of this paper is to present one-dimensional finitary linear cellular automata 𝑆 on Z m from an algebraic point of view. Among various other results, we (i) show that the Pontryagin dual S ̂ of 𝑆 is a classical one-dimensional linear cellular automaton 𝑇 on Z m ; (ii) give several equivalent conditions for 𝑆 to be invertible with inverse a finitary linear cellular automaton; (iii) compute the algebraic entropy of 𝑆, which coincides with the topological entropy of T = S ̂ by the so-called Bridge Theorem. In order to better understand and describe entropy, we introduce the degree deg ( S ) and deg ( T ) of 𝑆 and 𝑇.

Funding statement: The first author (H. Akın) was supported by the Simons Foundation (10.13039/100000893), United States and the Institute of International Education, United States.

Acknowledgements

It is a pleasure to thank the referee for the very careful reading and various helpful suggestions. The second and the third authors are members of the group GNSAGA of INdAM.

  1. Communicated by: Timothy C. Burness

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Received: 2023-06-20
Revised: 2023-11-01
Published Online: 2024-01-31

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