Some examples of cocycles with simple continuous singular spectrum (Q2717578)

This paper investigates the spectrum of a particular type of Anzai skew product. More specifically, the paper considers the automorphism on \(\mathbb{T}^2\) defined by \(T_\phi(z,w)= (e^{2\pi i\alpha}z, \phi(z)w)\), where \(\alpha\) is an irrational number between \(0\) and \(1\) whose continued fraction expansion \(\alpha= [0; a_1,a_2,\dots]\) has a subsequence \(n_k\) such that \(\lim_{k\to\infty} a_{n_k}= \infty\) and \(\phi\) is a function on the \(\{z\in\mathbb{C}:|z|= 1\}\) which is piecewise absolutely continuous and satisfies certain properties about its discontinuities. For such \(T_\phi\), the paper shows it has simple continuous singular spectrum on the complement of \(\{g: g(z, w)= f(z) w^m\), where \(f\) is an \(L^2\) function\}. The introduction summarizes the spectral properties of many known examples, and the proof involves many technical results involving \(\delta\)-weak mixing and various spectral conditions.