The topological entropy of invertible cellular automata

This paper is concerned with the topological entropy of invertible one-dimensional linear cellular automata, i.e., the maps T f [ - r , r ] : Z m Z → Z m Z which are given by T f [ - r , r ] ( x ) = ( 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 the ring Z m ( m ⩾ 2 ) by means of algorithm defined by D’amica et al. [On computing the entropy of cellular automa, Theoret. Comput. Sci. 290 (2003) 1629–1646]. We prove that if a one-dimensional linear cellular automata is invertible, then the topological entropies of this cellular automata and its inverse are equal.