The algebraic entropy of one-dimensional finitary linear cellular automata

Abstract

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