Cellular automata and Lyapunov exponents

The first definition of Lyapunov exponents (depending on a probability measure) for a one-dimensional cellular automaton were introduced by Shereshevsky in 1991. The existence of an almost everywhere constant value for each of the two exponents (left and right), requires particular conditions for the measure. Shereshevsky establishes an inequality involving these two constants and the metric entropies of both the shift and the cellular automaton. In this article we first prove that the two Shereshevsky’s exponents exist for a more suitable class of measures, then, keeping the same conditions, we define new exponents, called average Lyapunov exponents smaller or equal to the first ones. We obtain two inequalities: the first one is analogous to the Shereshevsky’s but concerns the average exponents; the second is the Shereshevsky inequality but with more suitable assumptions. These results are illustrated by two non-trivial examples, both proving that average exponents provide a better bound for the entropy, and one showing that the inequalities are strict in general.