The algebraic entropy of one-dimensional finitary linear cellular automata

2023/06/19 by Hasan Akın, Akın, Hasan, Dikran Dikranjan +5 · 1 citation
Computer Science · Mathematics · #20K30 #37B15 #37B40 #Cellular Automata and Applications #Computability, Logic, AI Algorithms #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR) #Mathematical Dynamics and Fractals #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2306.10838

openalex publication_date 2023/06/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The aim of this paper is to present one-dimensional finitary linear cellular automata S on \mathbb Z_m from an algebraic point of view. Among various other results, we: (i) show that the Pontryagin dual \widehat S of S is a classical one-dimensional linear cellular automaton T on \mathbb 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=\widehat S by the so-called Bridge Theorem. In order to better understand and describe the entropy we introduce the degree deg(S) and deg(T) of S and T.

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