确定性$[T,T^{-1}]$系统的同构与慢熵
Isomorphisms and slow entropy of deterministic $[T,T^{-1}]$ systems
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中文总结 AI 辅助
本文研究确定性$[T,T^{-1}]$系统的同构刚性，证明斜积同构条件，并构造任意低慢熵及违反慢熵变分原理的新例子。
中文摘要 AI 辅助
我们研究由无理旋转驱动的斜积保测系统，其中阶梯函数在圆的两半上分别取值$1$和$-1$。这些系统可视为$[T,T^{-1}]$系统的确定性版本，并且是Rokhlin余循环扩张的特殊实例。在遍历性假设下，我们证明这些系统具有有趣的刚性性质，即只可能存在显然的同构。我们证明两个斜积同构当且仅当底空间中的旋转同构，且纤维变换是翻转同构。在温和的额外假设下，我们能够完全刻画所有可能的同构映射。我们证明，通过在底空间中适当选择旋转角度，这些斜积可以实现任意低的下慢熵意义下的测度论复杂度。我们应用此结果获得违反慢熵变分原理的新系统例子。这些例子可以具有任意丰富的不变测度集合（任何可度量化Choquet单纯形，直至仿射同胚）。
英文摘要
We study skew product measure-preserving systems driven by an irrational rotation and the step function with values $1$ and $-1$ on each half of the circle. These systems can be seen as a deterministic version of Kalikow's $[T,T^{-1}]$ system, and are particular instances of Rokhlin cocycle extensions. Under the assumption of ergodicity, we show that these systems obey an interesting rigidity property for isomorphisms. That is, we show that two skew products are isomorphic if and only if the rotations in the base are isomorphic, and the fiber transformations are flip isomorphic. Under mild extra assumptions, we are able to characterize all isomorphisms. We prove that by choosing the rotation angle suitably, these systems can realize arbitrarily low measure-theoretic complexity in the sense of lower slow entropy. We apply this result to obtain new examples of systems that fail the variational principle for slow entropy. These examples can have a set of invariant measures as rich as desired (any metrizable Choquet simplex, up to affine homeomorphism).
发表机构
- Jagiellonian University（雅盖隆大学）
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