秩一系统、灵活类与 Shannon 轨道等价
Rank-one systems, flexible classes and Shannon orbit equivalence
AI总结：
本文引入秩一变换的灵活类，统一证明每个灵活类中都存在与通用里程计 Shannon 轨道等价的系统，并据此说明强混合性和特征值有理性质不是该等价下的不变量。
摘要：
我们在通用里程计与多种秩一系统之间建立了 Shannon 轨道等价。这一构造以统一方式完成，使用了我们所称的秩一变换的灵活类。我们的主要结果是：每个灵活类都包含一个与通用里程计 Shannon 轨道等价的元素。由于当 T 是里程计时，灵活类的一个典型例子是 {T}，因此我们的工作推广了 Kerr 和 Li 最近的一项结果，该结果表明每个里程计都与通用里程计 Shannon 轨道等价。当灵活类为单元素集时，主要结果给出的秩一变换是显式的。这适用于里程计和 Chacon 映射。我们还证明，强混合系统、具有给定特征值的系统，或角度属于实数轴任意固定开子集的无理旋转，都构成灵活类。特别地，强混合性以及特征值的有理性或无理性在 Shannon 轨道等价下均不保持。
英文摘要：
We build a Shannon orbit equivalence between the universal odometer and a variety of rank-one systems. This is done in a unified manner, using what we call flexible classes of rank-one transformations. Our main result is that every flexible class contains an element which is Shannon orbit equivalent to the universal odometer. Since a typical example of flexible class is $\{T\}$ when $T$ is an odometer, our work generalizes a recent result by Kerr and Li, stating that every odometer is Shannon orbit equivalent to the universal odometer. When the flexible class is a singleton, the rank-one transformation given by the main result is explicit. This applies to odometers and Chacon's map. We also prove that strongly mixing systems, systems with a given eigenvalue, or irrational rotations whose angle belongs to any fixed open subset of the real line form flexible classes. In particular, strong mixing, rationality or irrationality of the eigenvalues are not preserved under Shannon orbit equivalence.
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