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,T-r]:Z(m)(Z)-> Z(m)(Z) which are given by T-f[-r,T-r](x)=(y(n))(n)(infinity)=-infinity y(n)=f(x(n-r), ...,x(n+r))=Sigma(r)(i)=-r(n+i)(lambda ix) (mod m), x=(x(n))(n)(infinity)=-infinity is an element of Z(m)(z) and f: Z(m)(2r+1)-> Z(m), over the 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. (c) 2007 Elsevier B.V. All rights reserved.