Some examples of cocycles with simple continuous singular spectrum

K. Fr/aczek ¹

¹ Faculty of Mathematics and Computer Science Nicholas Copernicus University Chopina 12/18 87-100 Toru/n, Poland

Studia Mathematica, Tome 146 (2001) no. 1, pp. 1-13

Résumé

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.

Détail

DOI : 10.4064/sm146-1-1

Keywords: study spectral properties anzai skew products varphi mathbb rightarrow mathbb defined varphi omega alpha varphi omega where alpha irrational varphi mathbb rightarrow mathbb measurable cocycle precisely where varphi piecewise absolutely continuous sum jumps varphi equals zero shown simple continuous singular spectrum varphi orthocomplement space functions depending only first variable typical property above mentioned class cocycles alpha admits sufficiently fast approximation