Rescaled entropy of cellular automata

Abstract

For a $d-$dimensional cellular automaton with $d\geq1$ we introduce a rescaled entropy which estimates the growth rate of the entropy at small scales by generalizing previous approaches [1, 9]. We also define a notion of Lyapunov exponent and proves a Ruelle inequality as already established for $d = 1$. Finally we generalize the entropy formula for$1$-dimensional permutative cellular automata to the rescaled entropy in higher dimensions. This last result extends recent works of Shinoda and Tsukamoto dealing with the metric mean dimensions of two-dimensional symbolic dynamics.