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

In this paper we study the topological and metric directional entropy of ${\mathbb{Z}}^2$-actions by generated additive cellular automata (CA hereafter), defined by a local rule $f[l,r]$, $l,r\in \mathbb{Z}$, $l\le r$, i.e. the maps $T_{f[l,r]}:{\mathbb{Z}}_m^{\mathbb{Z}}\to {\mathbb{Z}}_m^{\mathbb{Z}}$ which are given by $T_{f[l,r]}(x)=(y_n{)}_{-\mathrm{\infty }}^{\mathrm{\infty }}$, $y_{n} = f(x_{n+l}, …, x_{n+r}) = \sum_{i=l}^r\lambda_{i}x_{i+n}(mod m)$,$x=(x_n)_ {n=-\infty}^{\infty}\in $\mathbb{Z}$^$\mathbb{Z}${m}$,and$f: $\mathbb{Z}${m}^{r-l+1}\to $\mathbb{Z}${m}$,overthering$$\mathbb{Z}$m (m \geq 2)$,andtheshiftmapactingoncompactmetricspace$$\mathbb{Z}$^$\mathbb{Z}${m}$,where$m$ $(m\ge 2)$ is a positive integer. Our main aim is to give an algorithm for computing the topological directional entropy of the ${\mathbb{Z}}^2$-actions generated by the additive CA and the shift map. Thus, we ask to give a closed formula for the topological directional entropy of ${\mathbb{Z}}^2$-action generated by the pair $(T{f[l, r]}, \sigma)$inthedirection$\theta$ 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 Ak\i 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.