The entropy and reversibility of cellular automata on Cayley tree

Abstract

In this paper, we study linear cellular automata (CAs) on Cayley tree of order 2 over the field $\mathbb F_p$ (the set of prime numbers modulo $p$). We construct the rule matrix corresponding to finite cellular automata on Cayley tree. Further, we analyze the reversibility problem of this cellular automaton for some given values of $a,b,c,d\in \mathbb{F}_{p}\setminus \{0\}$ and the levels $n$ of Cayley tree. We compute the measure-theoretical entropy of the cellular automata which we define on Cayley tree. We show that for CAs on Cayley tree the measure entropy with respect to uniform Bernoulli measure is infinity.