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In this paper, we first prove the strong Birkhoff Ergodic Theorem forsubharmonic functions with the irrational shift on the Torus. Then, it isapplied to the analytic quasi-periodic Jacobi cocycles. We show that if theLyapunov exponent of these cocycles is positive at one point, then it ispositive on an interval centered at this point for suitable frequency andcoupling numbers. We also prove that the Lyapunov exponent is H\"oldercontinuous 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 theLyapunov exponent is always positive for all irrational frequencies andH\"older continuous in $E$ for all finite Liouville frequencies. We also studythe Lyapunov exponent of the Schr\"odinger cocycles, a special case of theJacobi ones, and obtain its H\"older continuity in the frequency.

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