In this talk, we investigate large deviation bounds for the measure of deviation sets associated with asymptotically additive and sub-additive potentials under some weak specification properties. As an application of the large deviation results, we obtain exponential large deviation bounds in the Shannon-McMillan-Breiman convergence formula for entropy in the case of weak Gibbs measures and topologically mixing subshifts of nite type. Other applications include  large deviation estimates for Lyapunov exponents and pointwise dimension. This is a joint work with Paulo Varandas.

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