Synchronization and maximum Lyapunov exponents of cellular automata

We study the synchronization of totalistic one-dimensional cellular automat s (CA). The CA with a nonzero synchronization threshold exhibit complex non periodic space time patterns and vice versa. This synchronization transitio n is related to directed percolation. We also study the maximum Lyapunov ex ponent for CA, defined analogous to continuous dynamical systems as the exp onential rate of expansion of the linear map induced by the evolution rule of CA, constructed with the aid of the Boolean derivatives. The synchroniza tion threshold is strongly correlated to the maximum Lyapunov exponent and we propose approximate relations between these quantities. The value of thi s threshold can he used to parametrize the space time complexity of CA. [S1 063-651X(99)51502-X].