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

TL;DR

This paper develops an algorithm to compute the topological directional entropy of $Z^2$-actions generated by linear cellular automata, providing a closed-form formula based on local rule coefficients, extending previous results to invertible linear CA.

Contribution

The paper introduces a method to calculate the topological directional entropy for $Z^2$-actions generated by linear cellular automata, generalizing prior work to all invertible linear CA.

Findings

Provides an explicit algorithm for entropy calculation.

Derives a closed-form formula based on local rule coefficients.

Extends previous results to all invertible 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 $f[l, r]$, $l, r\in \mathbb{Z}$, $l\leq r$, i.e. the maps $T_{f[l, r]}: \mathbb{Z}^\mathbb{Z}_{m} \to \mathbb{Z}^\mathbb{Z}_{m}$ which are given by $T_{f[l, r]}(x) =(y_n)_ {-\infty}^{\infty}$, $y_{n} = f(x_{n+l}, ..., x_{n+r}) = \sum_{i=l}^r\lambda_{i}x_{i+n}(mod m)$, $x=(x_n)_ {n=-\infty}^{\infty}\in \mathbb{Z}^\mathbb{Z}_{m}$, and $f: \mathbb{Z}_{m}^{r-l+1}\to \mathbb{Z}_{m}$, over the ring $\mathbb{Z}_m (m \geq 2)$, and the shift map acting on compact metric space $\mathbb{Z}^\mathbb{Z}_{m}$, where $m$ $(m \geq2)$ 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 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.