The Measure-Theoretical Entropy of a Linear Cellular Automata with respect to a Markov Measure

Abstract

In this paper we study the measure-theoretical entropy of the one-dimensional linear cellular automata (CA hereafter) $T_{f[-l,r]}$, generated by local rule $f(x_{-l},...,x_r)=\sum _{i=-l}^r{\lambda }_i{x}_i(\text{mod}m)$, where $l$ and $r$ are positive integers, acting on the space of all doubly infinite sequences with values in a finite ring ${\mathbb{Z}}_m$, $m\ge 2$, with respect to a Markov measure. We prove that if the local rule $f$ is bipermutative, then the measure-theoretical entropy of linear CA $T_{f[-l,r]}$ with respect to a Markov measure ${\mu }_{\pi P}$ is $h_{{\mu }_{\pi P}}(T_{f[-l,r]})=-(l+r)\sum _{i,j=0}^{m-1}{p}_i{p}_{ij}\text{log}{p}_{ij}.$

7 pages