The algebraic entropy of one-dimensional finitary linear cellular automata

The aim of this paper is to present one-dimensional finitary linear cellular automata S on Zm 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 Zm; (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.