The topological entropy of nth iteration of an additive cellular automata

Abstract

In this paper, our aim is to investigate the topological entropy of nth iteration of an one-dimensional additive cellular automata (CA hereafter), i.e. the maps T : Z m Z → Z m Z which are given by Tx = ( y n ) n = - ∞ ∞ , y n = F ( x n - r , … , x n + r ) = ∑ i = - r r λ i x n + i ( mod m ) , x = ( x n ) n = - ∞ ∞ ∈ Z m Z and F : Z m 2 r + 1 → Z m , over Z m ( m ⩾ 2 ) by means of both the algorithm and Lyapunov exponents of the CA T that is given by D’amico et al. [Theor. Comput. Sci. 290 (2003) 1629–1646]. We show that if the local rule F is bipermutative (in Hedlund’s terminology), then the topological entropy of nth iteration of one-dimensional additive CA is 2 nr log m. We obtain necessary and sufficient conditions for the topological entropy of nth iteration of CA to be 2 nr log m. We show that the uniform Bernoulli measure is a measure of maximal entropy for the nth iteration of the CA generated by bipermutative local rule F.