In this paper we consider the discrete dynamical systemwhereis a stationary ergodic Markov chain with state spaceis a bounded C1map, andWe first give conditions under which the Lyapunov spectrum associated with this dynamical system reduces, with probability one, to the top Lyapunov exponent. We then investigate the relationship between this exponent and the moment Lyapunov exponents of the system and give a basic large deviation result