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Hölder continuity of the Lyapunov exponent for analytic quasiperiodic Schrödinger cocycle with weak Liouville frequency. (English) Zbl 1315.39004

Authors’ abstract: For analytic quasiperiodic Schrödinger cocycles \({(\omega, A)}\), where \({\omega\in\mathbb{R}\setminus\mathbb{Q}}\) and \[ A=A(x,E)=\left(\begin{matrix} E-v(x) & -1\\ 1& 0\\ \end{matrix}\right),\;E\in\mathbb{R}, \,v:\mathbb{T}\to\mathbb{R}\;\text{is real analytic}, \] M. Goldstein and W. Schlag [Ann. Math. (2) 154, No. 1, 155–203 (2001; Zbl 0990.39014)] proved that the Lyapunov exponent \[ L(\omega, E)=\lim_{n\to\infty}\frac{1}{n}\int_{\mathbb{T}}\left\|\prod_{k=n}^1A(x+k\omega,E)\right\|dx \] is Hölder continuous provided that the base frequency \(\omega\) satisfies a strong Diophantine condition. In this paper, we give a refined large deviation theorem, which implies the Hölder continuity of the Lyapunov exponent for all Diophantine frequencies \(\omega\), even for weak Liouville \(\omega\), which improves the result of Goldstein and Schlag [loc. cit.].

MSC:

39A12 Discrete version of topics in analysis
37H15 Random dynamical systems aspects of multiplicative ergodic theory, Lyapunov exponents
26A16 Lipschitz (Hölder) classes

Citations:

Zbl 0990.39014
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References:

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