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## Explicit rank-1 constructions for irrational rotations

Studia Mathematica MSC: Primary 37A05; Secondary 37A20, 37A40. DOI: 10.4064/sm220214-13-10 Opublikowany online: 1 December 2022

#### Streszczenie

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.

#### Autorzy

• Alexandre I. DanilenkoB. Verkin Institute for Low Temperature Physics & Engineering