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

Consider the cellular automata (CA) of ${\mathbb{Z}}^2$-action $\mathrm{\Phi }$ on the space of all doubly infinite sequences with values in a finite set ${\mathbb{Z}}_r$, $r\ge 2$ determined by cellular automata $T_{F[-k,k]}$ with an additive automaton rule $F(x_{n-k},\mathrm{.}\mathrm{.}\mathrm{.},x_{n+k})={\sum }_{i=-k}^k{a}_i{x}_{n+i}(modr)$. It is investigated the concept of the measure theoretic directional entropy per unit of length in the direction ${\omega }_0$. It is shown that $h_{\mu }(T_{F[-k,k]}^u)=u{h}_{\mu }(T_{F[-k,k]})$, $h_{\mu }({\mathrm{\Phi }}^u)=u{h}_{\mu }(\mathrm{\Phi })$ and $h_{\vec{v}}({\mathrm{\Phi }}^u)=u{h}_{\vec{v}}(\mathrm{\Phi })$ for $\vec{v}\in {\mathbb{Z}}^2$ where $h$ is the measure-theoretic entropy.