The Measure-theoretical Entropy of a Linear Cellular Automata with Respect to a Markov Measure

In this paper we study the measure-theoretical entropy of the one-dimensional linear cellular automata (CA hereafter) Tf [−l,r], generated by local rule f(x −l, . . . , xr) = r ∑ i=−l λixi(mod m), where l and r are positive integers, acting on the space of all doubly infinite sequences with values in a finite ring Zm, m ≥ 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 Tf [−l,r] with respect to a Markov measure μπP is hμπP (Tf [−l,r]) = −(l+ r) m−1 ∑ i,j=0 pipij log pij .