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, we apply it to the analytic quasi-periodic Jacobi cocycles and show that for suitable frequency and coupling number, if the Lyapunov exponent of these cocycles is positive at one point, then it is positive on an interval centered at this point and Holder continuous in E on this interval. What's more, if the coupling number of the potential is large, then the Lyapunov exponent is always positive for all irrational frequencies and Holder continuous in E for all finite Liouville frequencies. For the Schrodinger cocycles, a special case of the Jacobi ones, its Lyapunov exponent is also Holder continuous in the frequency and the lengths of the intervals where the Holder condition of the Lyapunov exponent holds only depend on the coupling number.