Abstract
Abstract We show that for an additive one-dimensional cellular automaton (CA) Z 2 -action Φ on the space of all doubly infinite sequences with values in a finite set Z a = { 0 , 1 , 2 , … , a - 1 } , determined by an additive automaton rule F ( x n - k , … , x n + k ) = ∑ i = - k k λ i x n + i ( mod a ) , and a Φ -invariant uniform Bernoulli measure, the directional entropy is h v → ( Φ ) = 2 k 2 log a for v → = ( k 1 , k 2 ) ∈ Z + 2 , where k 2 > k 1 and λ i ∈ Z a . 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 .
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