Strong Birkhoff Ergodic Theorem for subharmonic functions with irrational shift and its application to analytic quasi-periodic cocycles

In this paper, we first prove the strong Birkhoff Ergodic Theorem for subharmonic functions with the irrational shift on the Torus. Then, it is applied to the analytic quasi-periodic Jacobi cocycles. We show that if the Lyapunov exponent of these cocycles is positive at one point, then it is positive on an interval centered at this point for suitable frequency and coupling numbers. We also prove that the Lyapunov exponent is H\"older continuous in $E$ on this interval and calculate the expression of its length. What's more, if the coupling number of the potential is large, then the Lyapunov exponent is always positive for all irrational frequencies and H\"older continuous in $E$ for all finite Liouville frequencies. We also study the Lyapunov exponent of the Schr\"odinger cocycles, a special case of the Jacobi ones, and obtain its H\"older continuity in the frequency.