Rescaled entropy of cellular automata

For a d-dimensional cellular automaton with d $\ge$ 1 we introduce rescaled entropy which estimates the growth rate of at small scales by generalizing previous approaches [1, 9]. We also define notion Lyapunov exponent and proves Ruelle inequality as already established for = in [16, 15]. Finally generalize formula 1-dimensional permutative automata [18] to higher dimensions. This last result extends recent works [17] Shinoda Tsukamoto dealing metric mean dimensions two-dimensional symbolic dynamics.