On the topological directional entropy
Journal of Computational and Applied Mathematics
In this paper we study the topological and metric directional entropy of Z^2-actions by generated additive cellular automata (CA hereafter), defined by a local rule f[l, r], l, r∈Z, l≤ r, i.e. the maps T_f[l, r]: Z^Z_m→Z^Z_m which are given by T_f[l, r](x) =(y_n)_ -∞^∞, y_n = f(x_n+l, ..., x_n+r) = ∑_i=l^rλ_ix_i+n(mod m), x=(x_n)_ n=-∞^∞∈Z^Z_m, and f: Z_m^r-l+1→Z_m, over the ring Z_m (m ≥ 2), and the shift map acting on compact metric space Z^Z_m, where m (m ≥2) is a positive integer. Our main
... im is to give an algorithm for computing the topological directional entropy of the Z^2-actions generated by the additive CA and the shift map. Thus, we ask to give a closed formula for the topological directional entropy of Z^2-action generated by the pair (T_f[l, r], σ) in the direction θ that can be efficiently and rightly computed by means of the coefficients of the local rule f as similar to [Theor. Comput. Sci. 290 (2003) 1629-1646]. We generalize the results obtained by Akı n [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.