  In this paper we introduce a notion of free energy and large deviations rate
function for asymptotically additive sequences of potentials via an
approximation method by families of continuous potentials. We provide estimates
for the topological pressure of the set of points whose non-additive sequences
are far from the limit described through Kingman's sub-additive ergodic theorem
and give some applications in the context of Lyapunov exponents for
diffeomorphisms and cocycles, and Shannon-McMillan-Breiman theorem for Gibbs
measures.
