# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2021162
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## Strong Birkhoff ergodic theorem for subharmonic functions with irrational shift and its application to analytic quasi-periodic cocycles

 College of Sciences, Hohai University, No.1 Xikang Road, Nanjing, Jiangsu 210098, China

* Corresponding author: Kai Tao

Received  February 2020 Revised  August 2020 Early access November 2021

Fund Project: The author is supported by The author is supported by the Fundamental Research Funds for the Central Universities (Grant B200202004) and China Postdoctoral Science Foundation (Grant 2019M650094)

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 Hölder 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 Hölder continuous in $E$ for all finite Liouville frequencies. For the Schrödinger cocycles, a special case of the Jacobi ones, its Lyapunov exponent is also Hölder continuous in the frequency and the lengths of the intervals where the Hölder condition of the Lyapunov exponent holds only depend on the coupling number.

Citation: Kai Tao. Strong Birkhoff ergodic theorem for subharmonic functions with irrational shift and its application to analytic quasi-periodic cocycles. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021162
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