On Directional Entropy of a $\mathbb{Z}^{2}$-Action

Abstract

Consider the cellular automata (CA) of $\mathbb{Z}^{2}$-action $\Phi$ on the space of all doubly infinite sequences with values in a finite set $\mathbb{Z}_{r}$, $r \geq 2$ determined by cellular automata $T_{F[-k, k]}$ with an additive automaton rule $F(x_{n-k},...,x_{n+k})=\sum\limits_{i=-k}^{k}a_{i}x_{n+i}(mod r)$. It is investigated the concept of the measure theoretic directional entropy per unit of length in the direction $\omega_{0}$. It is shown that $h_{\mu}(T_{F[-k,k]}^{u})=uh_{\mu}(T_{F[-k,k]})$, $h_{\mu}(\Phi^{u})=uh_{\mu}(\Phi)$ and $h_{\vec{v}}(\Phi^{u})=uh_{\vec{v}}(\Phi)$ for $\vec{v} \in \mathbb{Z}^{2}$ where $h$ is the measure-theoretic entropy.