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Almost everywhere positivity of the Lyapunov exponent for the doubling map. (English) Zbl 1087.47037

This article deals with the discrete Schrödinger operator \[ [H_\theta\phi](n) = \phi(n+1) + \phi(n-1) + \lambda f(2^n\theta)\phi(n) \] on the space \(\ell^2({\mathbb Z}_+)\), with Dirichlet boundary condition \(\phi(0) = 0\); it is assumed that \(\lambda > 0\), \(\theta \in {\mathbb T} = {\mathbb R} / {\mathbb Z}\), \(f: \;{\mathbb T} \to {\mathbb R}\) is measurable, bounded and non-constant. Under these assumptions, there exists a function (the Lyapunov exponent) \[ \gamma(E) = \lim_{n \to \infty} \;\frac1n \log \| M(n,E,\theta)\| \;\;\text{for almost every} \;\;\theta, \] where \(M(nE,\theta)\) is the transfer matrix from \(1\) to \(n\) for the operator \(H_\theta\) at energy \(E \in {\mathbb R}\): \[ M(n,E,\theta) = \begin{pmatrix} E - \lambda f(2^n\theta) & -1 \\ 1 & 0\end{pmatrix} \times \dots \times \begin{pmatrix} E - \lambda f(2^n\theta) & -1 \\ 1 & 0\end{pmatrix}. \] The authors prove that the Lyapunov exponent \(\gamma(E)\) is positive for almost every \(E \in {\mathbb R}\) and the absolutely continuous spectrum of \(H_\theta\) is empty for almost every \(\theta \in {\mathbb T}\). Some possible generalizations are also mentioned.

MSC:

47B39 Linear difference operators
47B36 Jacobi (tridiagonal) operators (matrices) and generalizations
39A70 Difference operators
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