Rescaled entropy of cellular automata

TL;DR

This paper introduces a rescaled entropy for d-dimensional cellular automata to analyze their growth rates, extends the concept of Lyapunov exponents, and generalizes entropy formulas from 1D to higher dimensions, linking to recent symbolic dynamics research.

Contribution

It defines a new rescaled entropy for cellular automata in higher dimensions and extends existing entropy and Lyapunov exponent results to these cases.

Findings

Introduced a rescaled entropy estimating growth rates at small scales.

Proved a Ruelle inequality for higher-dimensional cellular automata.

Generalized entropy formulas from 1D to higher dimensions.

Abstract

For a d-dimensional cellular automaton with d $\ge$ 1 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 in [16, 15]. Finally we generalize the entropy formula for 1-dimensional permutative cellular automata [18] 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.