On computing the Lyapunov exponents of reversible cellular automata

Abstract: We consider the problem of computing the Lyapunov exponents of reversible cellular automata (CA). We show that the class of reversible CA with right Lyapunov exponent $2$ cannot be separated algorithmically from the class of reversible CA whose right Lyapunov exponents are at most $2-\delta$ for some absolute constant $\delta>0$. Therefore there is no algorithm that, given as an input a description of an arbitrary reversible CA $F$ and a positive rational number $\epsilon>0$, outputs the Lyapunov exponents of $F$ with accuracy $\epsilon$. We also compute the average Lyapunov exponents (with respect to the uniform measure) of the CA that perform multiplication by $p$ in base $pq$ for coprime $p,q>1$.