The Topological Directional Entropy of 2 - actions Generated by Linear Cellular Automata

In this paper we study the topological directional entropy of 2  -actions by generated linear cellular automata (CA hereafter), defined by a local rule f[l,r], l, r , l  r, i.e. the maps Tf[l, r]: m m      which are given by Tf[l, r](x) = ( ) n n y   , yn = f(xn+l, ..., xn+r) = r x i n i i l     (mod m), x=    n n x ) (  m   and f: 1 r l m     Zm, over the ring Zm (m  2), and the shift map acting on compact metric space m   . We give a closed formula, which can be efficiently and rightly computed by means of the coefficients of the local rule f, for the topological directional entropy of 2  -action generated by the pair (Tf[l, r], ) in the direction θ. We generalize the results obtained by Akın [The topological entropy of invertible cellular automata, J. Comput. Appl. Math. 213 (2) (2008) 501 – 508] to the topological entropy of any invertible linear CA.