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 Tf[-r,r]:ZmZ→ZmZ which are given by Tf[-r,r](x)=(yn)n=-∞∞, yn=f(xn-r,…,xn+r)=∑i=-rrλixn+i(modm), x=(xn)n=-∞∞∈ZmZ and f:Zm2r+1→Zm, over the ring Zm(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.