Explicit rank-one constructions for irrational rotations

TL;DR

This paper explicitly constructs rank-one transformations for irrational rotations, providing new examples with specific spectral properties and invariant measures, solving a problem posed by del Junco.

Contribution

It offers explicit rank-one constructions for all irrational rotations, including well approximable ones, with detailed spectral and measure-theoretic properties.

Findings

Constructed rank-one systems with eigenvalues for all irrationals.

Explicit invariant measures for rigid and zero type systems.

Identified systems with trivial centralizer for zero type case.

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 $\Bbb T$. 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 $\mu_{\theta}'$ on $\Bbb T$ are constructed explicitly such that $(\Bbb T,\mu_\theta, R_\theta)$ is {\it rigid} and of rank one and $(\Bbb T,\mu_\theta', R_\theta)$ 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.