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Abstract

Let $\theta \in (0,1)$ be an irrational number and let $\lambda :=e^{2\pi i\theta }$. For each well approximable irrational $\theta $, we provide an explicit rank-1 construction of the $\lambda $-rotation $R_\lambda $ on the circle $\Bbb T$. This solves “almost surely” a problem by del Junco. For every irrational $\theta $, we construct explicitly a rank-1 transformation with an eigenvalue $\lambda $. For every irrational $\theta $, two infinite $\sigma $-finite invariant measures $\mu _\lambda $ and $\mu _{\lambda }’$ on $\Bbb T$ are constructed explicitly such that $(\Bbb T,\mu _\lambda , R_\lambda )$ is {rigid} and of rank 1 and $(\Bbb T,\mu _\lambda ’, R_\lambda )$ is of zero type and of rank 1. The centralizer of the latter system consists of just the powers of $R_\lambda $. Some versions of the aforementioned results are proved under an extra condition on boundedness of the sequence of cuts in the rank-1 construction.