可计算保测变换的有效回归性
Effective recurrence for computable measure-preserving transformations
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中文总结 AI 辅助
本文研究了可计算保测变换下庞加莱回归的有效条件，提出了$\Pi^0_n$-生成性等新概念，并给出了正频率回归的充分必要条件。
中文摘要 AI 辅助
我们证明了若干充分必要条件，在这些条件下，一个点对于所有可计算（遍历）保测变换以及所有特定复杂度的集合满足庞加莱回归定理。必要条件通过构造违反回归性的特定保测变换而获得。虽然其中一些条件涉及算法随机性的标准概念，其他条件则涉及在可计算概率空间背景下发展的新的生成性概念，我们称之为$\Pi^0_n$-生成性和拟$\Pi^0_n$-生成性。我们还提供了点在包含它们的集合中以正频率回归的条件，涉及简单回归和多重回归。
英文摘要
We prove several necessary and sufficient conditions under which a point satisfies the Poincaré Recurrence Theorem for all computable (ergodic) measure-preserving transformations and all sets of a particular complexity. The necessary conditions are obtained by constructing specific measure-preserving transformations which violate recurrence. While some of these conditions pertain to standard notions of algorithmic randomness, others involve new notions of genericity developed in the context of a computable probability space, which we call $Π^0_n$-genericity and quasi-$Π^0_n$-genericity. We also provide conditions under which points recur at a positive frequency in sets containing them, regarding both simple and multiple recurrence.
发表机构
- Penn State University（宾夕法尼亚州立大学）
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