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

Abstract

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 $T_{f[l,r]}(x)=(y_n{)}_{-\infty }^{\infty }$, $y_n=f(x_{n+l},...,x_{n+r})={\sum }_{i=l}^r{\lambda }_i{x}_{i+n}(modm)$, $x=(x_n{)}_{n=-\infty }^{\infty }\in {\mathbb{Z}}_m^{\mathbb{Z}}$, and $f:{\mathbb{Z}}_m^{r-l+1}\to {\mathbb{Z}}_m$, over the ring ${\mathbb{Z}}_m(m\ge 2)$, and the shift map acting on compact metric space ${\mathbb{Z}}_m^{\mathbb{Z}}$, 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 $(T_{f[l,r]},\sigma )$ in the direction $\theta$ 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 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.

9 pages. submitted

References in corpus (1)

• Some Ergodic Properties of Invertible Cellular Automata