On the directional entropy of Z2-actions generated by additive cellular automata

摘要

We show that for an additive one-dimensional cellular automaton (CA) Z2-action Φ on the space of all doubly infinite sequences with values in a finite set Za={0,1,2,…,a-1}, determined by an additive automaton rule F(xn-k,…,xn+k)=∑i=-kkλixn+i(moda), and a Φ-invariant uniform Bernoulli measure, the directional entropy is hv→(Φ)=2k2loga for v→=(k1,k2)∈Z+2, where k2 > k1 and λi∈Za.We also calculate the measure-theoretic entropy for additive CA Z×N-actions Φ generated by some additive one-dimensional cellular automata and shift transformation and we show that uniform Bernoulli measure is a maximal measure for additive CA Z×N-actions Φ.