Open AccessStrong Birkhoff ergodic theorem for subharmonic functions with irrational shift and its application to analytic quasi-periodic cocycles
Author(s) -
Kai Tao
Publication year - 2022
Publication title -
discrete and continuous dynamical systems
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.289
H-Index - 70
eISSN - 1553-5231
pISSN - 1078-0947
DOI - 10.3934/dcds.2021162
Subject(s) - lyapunov exponent , ergodic theory , interval (graph theory) , mathematics , irrational number , exponent , torus , pure mathematics , mathematical analysis , combinatorics , physics , quantum mechanics , nonlinear system , geometry , linguistics , philosophy
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 \begin{document}$ E $\end{document} on this interval. What's more, if the coupling number of the potential is large, then the Lyapunov exponent is always positive for all irrational frequencies and Hölder continuous in \begin{document}$ E $\end{document} 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.