Rescaled entropy of cellular automata

For a d-dimensional cellular automaton with d 1 we introduce a rescaled entropy which estimates the growth rate of the entropy at small scales by generalizing previous approaches [1, 7]. We also define a notion of Lyapunov exponent and proves a Ruelle inequality as already established for d = 1 in [16, 18]. Finally we generalize the entropy formula for one-dimensional permutative cellular automata [19] to the rescaled entropy in higher dimensions. This last result extends recent works [17] of Shinoda and Tsukamoto dealing with the metric mean dimensions of two-dimensional symbolic dynamics.