Some examples of cocycles with simple continuous singular spectrum
Tom 146 / 2001
Studia Mathematica 146 (2001), 1-13
MSC: Primary 37A05.
DOI: 10.4064/sm146-1-1
Streszczenie
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.
