The algebraic entropy of one-dimensional finitary linear cellular automata (Q6565830)

Let \(A\) be a non-empty set (called the alphabet), \(G\) a group and let \(A^{G}=\{ x: G \rightarrow A \mid x \mbox{ map}\}\) be the set of configurations. For \(g \in G\), let \(L_{g}: G \rightarrow G\) be the left multiplication by \(g\). A map \(T : A^{G} \rightarrow A^{G}\) is a cellular automaton if there exist a finite subset \(M\) of \(G\) and a map \(f: A^{M}M \rightarrow A\) such that, for every \(x\in A^{G}\) and \(g \in G\), \(T(x)(g)=f((x \circ L_{g}) \upharpoonright_{M})\).\N\NThe aim of the paper is to present one-dimensional finitary linear cellular automata \(S\) on \(\mathbb{Z}_{m}\) from an algebraic point of view. First, the authors show that the Pontryagin dual \(\widehat{S}\) of \(S\) is a classical one-dimensional linear cellular automaton \(T\) on \(\mathbb{Z}_{m}\). Secondly, they give several equivalent conditions for \(S\) to be invertible with inverse a finitary linear cellular automaton. Finally, they compute the algebraic entropy of \(S\), which coincides with the topological entropy of \(T=\widehat{S}\) by the so-called bridge theorem (see the works of the second and third author [Topology Appl. 159, No. 13, 2980--2989 (2012; Zbl 1256.54061); ibid. 169, 21--32 (2014; Zbl 1322.37007)]).