Positive Lyapunov exponents for quasiperiodic Szegő cocycles (Q2895589)

The paper is devoted to the study of analytic quasi-periodic Szegő and Schrödinger cocycles, i.e., dynamical systems of the form NEWLINE\[NEWLINE(T,A): X \times \mathbb{C}^2 \to X \times \mathbb{C}^2,\quad (x,w) \to (R_\alpha(x), A(x)w),NEWLINE\]NEWLINE where \(X = \mathbb{R}/\mathbb{Z},\) \(R_\alpha(x)\) is the translation \(x\mapsto x + \alpha\) and \(A\) is either the Schrödinger cocycle map \(A^{(E-\lambda v)}(x): X \to\operatorname{SL}(2,\mathbb{R}),\) NEWLINE\[NEWLINEA^{(E-\lambda v)}(x) = \left( \begin{matrix} E-\lambda v(x) & -1 \\ 1 & 0 \end{matrix} \right)NEWLINE\]NEWLINE (\(E \in \mathbb{R},\) \(v\) is a real analytic potential) or the Szegő cocycle map \(A^{(E,f)}: X \to\operatorname{SU}(1,1),\) NEWLINE\[NEWLINEA^{(E,f)}(x) = (1 - |f(x)|^2)^{-1/2} \left( \begin{matrix} \sqrt{E} & -\overline{f(x)}/\sqrt{E} \\ -f(x)\sqrt{E} & 1/\sqrt{E} \end{matrix} \right),NEWLINE\]NEWLINE where \(E \in \partial \mathbb{D},\) \(\mathbb{D}\) is the open disc in \(\mathbb{C},\) and the measurable function \(f: X \to \mathbb{D}\) satisfies \(\int_X \ln(1 - |f|)d\mu > -\infty.\)NEWLINENEWLINEThe author constructs a class of analytic quasiperiodic Szegő cocycles with uniformly positive Lyapunov exponents and estimates the positive measure of \(E\) such that the cocycle \((R_\alpha, A^{(E,f)})\) is not uniformly hyperbolic (if \(\alpha\) is a Brjuno number and \(|f| \to 1\)).NEWLINENEWLINEAnalogous results are proved for Schrödinger cocycles with nonconstant real analytic potentials.