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

Kai Tao

Published 2018-05-01Version 1

In this paper, we 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 operators. We show that if the Lyapunov exponent is positive at one point, then it is positive on an interval centered at this point for suitable frequencies 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 any irrational frequency and H\"older continuous in $E$ for all Diophantine and some Liouville frequencies. We also study the Lyapunov exponent of the Schr\"odinger operators, a special case of the Jacobi ones, and obtain its H\"older continuity in the frequency.