Entropy of Multidimensional Cellular Automata

Since the topological entropy of a vast class of two-dimensional cellular automata (CA) is infinite, of interest is the possibility to renormalize it so that to obtain a positive finite value. We find the asymptotics of the information function of a multidimensional CA and, accordingly, introduce the renormalized topological entropy as a coefficient of this asymptotics. We describe some properties of the introduced quantity, in particular, its positivity for CA of the type of “The Game of Life.” Also, we give an example of an explicit evaluation of this parameter for a particular cellular automaton.