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

Kai Tao

doi:10.3934/dcds.2021162

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2022, Volume 42, Issue 3: 1495-1533. Doi: 10.3934/dcds.2021162

This issue Previous Article An Erratum on "Stability and dynamics of a weak viscoelastic system with memory and nonlinear time-varying delay" (Discrete Continuous Dynamic Systems, 40(3), 2020, 1493-1515) Next Article On long-time asymptotic behavior for solutions to 2D temperature-dependent tropical climate model

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

Kai Tao^,

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

^* Corresponding author: Kai Tao
^* Corresponding author: Kai Tao

Received: January 31, 2020

Revised: July 31, 2020

Early access: November 2021

Published: March 2022

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)

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Abstract

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.

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Mathematics Subject Classification: Primary: 37C55, 37F10, 37C40.

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