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Abstract

We study spectral properties of Anzai skew products $T_{\varphi }:{\mathbb T}^2\rightarrow {\mathbb T}^2$ defined by $$T_{\varphi }(z,\omega )=(e^{2\pi i\alpha }z,\varphi (z) \omega ),$$ where $\alpha $ is irrational and $\varphi :{\mathbb T}\rightarrow {\mathbb T}$ is a measurable cocycle. Precisely, we deal with the case where $\varphi $ is piecewise absolutely continuous such that the sum of all jumps of $\varphi $ equals zero. It is shown that the simple continuous singular spectrum of $T_{\varphi }$ on the orthocomplement of the space of functions depending only on the first variable is a “typical” property in the above-mentioned class of cocycles, if $\alpha $ admits a sufficiently fast approximation.