Zero temperature limits of equilibrium states for subadditive potentials and approximation of the maximal Lyapunov exponent

In this paper we study ergodic optimization problems for subadditive sequences of functions on a topological dynamical system. We prove that for $t\to \mathrm{\infty }$ any accumulation point of a family of equilibrium states is a maximizing measure. We show that the Lyapunov exponent and entropy of equilibrium states converge in the limit $t\to \mathrm{\infty }$ to the maximum Lyapunov exponent and entropy of maximizing measures. In the particular case of matrix cocycles we prove that the maximal Lyapunov exponent can be approximated by Lyapunov exponents of periodic trajectories under certain assumptions.