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arXiv 2111.07375math.DS
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无理旋转的显式秩一构造

Explicit rank-one constructions for irrational rotations

Alexandre I. Danilenko, Mykyta I. Vieprik

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

对于每个{ \it良好逼近}无理$θ$，我们提供了一个明确的秩1结构的$e^{2πiθ}$旋转$R_θ$上的圆$\Bbb T$。这“几乎肯定”解决了德尔·容克的一个问题。对于{ \it every}无理$θ$，我们构造了一个具有特征值$e^{2πiθ}$的秩一变换.对任意无理$θ$，构造了$\Bbb T$上的两个无穷$σ$ -有限不变测度$μ_θ$和$μ_θ'$，使得$(\Bbb T,μ_θ, R_θ)$是{ \it刚性}秩1的，$(\Bbb T,μ_θ', R_θ)$是{ \it零型}秩1的.后一系统的中心化器仅由$R_θ$的幂组成。在秩一构造的割序列有界性的一个附加条件下，证明了上述结果的某些版本。

英文摘要：

For each {\it well approximable} irrational $θ$, we provide an explicit rank-one construction of the $e^{2πiθ}$-rotation $R_θ$ on the circle $\Bbb T$. This solves "almost surely" a problem by del Junco. For {\it every} irrational $θ$, we construct explicitly a rank-one transformation with an eigenvalue $e^{2πiθ}$. For every irrational $θ$, two infinite $σ$-finite invariant measures $μ_θ$ and $μ_θ'$ on $\Bbb T$ are constructed explicitly such that $(\Bbb T,μ_θ, R_θ)$ is {\it rigid} and of rank one and $(\Bbb T,μ_θ', R_θ)$ is of {\it zero type} and of rank one. The centralizer of the latter system consists of just the powers of $R_θ$. Some versions of the aforementioned results are proved under an extra condition on boundedness of the sequence of cuts in the rank-one construction.

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