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

7 pages, submitted