Explicit rank-one constructions for irrational rotations

Abstract

For each {\it well approximable} irrational $\theta$, we provide an explicit rank-one construction of the $e^{2\pi i\theta }$-rotation $R_{\theta }$ on the circle $\mathbb{T}$. This solves "almost surely" a problem by del Junco. For {\it every} irrational $\theta$, we construct explicitly a rank-one transformation with an eigenvalue $e^{2\pi i\theta }$. For every irrational $\theta$, two infinite $\sigma$-finite invariant measures ${\mu }_{\theta }$ and ${\mu }_{\theta }^{\prime }$ on $\mathbb{T}$ are constructed explicitly such that $(\mathbb{T},{\mu }_{\theta },R_{\theta })$ is {\it rigid} and of rank one and $(\mathbb{T},{\mu }_{\theta }^{\prime },R_{\theta })$ is of {\it zero type} and of rank one. The centralizer of the latter system consists of just the powers of $R_{\theta }$. Some versions of the aforementioned results are proved under an extra condition on boundedness of the sequence of cuts in the rank-one construction.

A transliteration flaw in the name of the second named author is removed