保测流的别林斯卡娅定理的类似物
The analogue of Belinskaya's theorem for measure-preserving flows
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中文总结 AI 辅助
研究保测流的别林斯卡娅定理类似物，证明\(\mathrm{L}^1\)全群同构的自由遍历保测流在标量时间变化下共轭，回答相关问题，关键新要素是可公度准则，由双智能体AI系统发现，作者验证证明。
中文摘要 AI 辅助
我们证明了保测流的别林斯卡娅定理的类似物：两个自由遍历保测流，其\(\mathrm{L}^1\)全群作为抽象群同构，则它们在标量时间变化下共轭。这回答了弗朗索瓦·勒梅特和作者提出的一个问题。我们表明，只要两个自由遍历流生成相同的轨道等价关系，且其中一个包含在另一个的\(\mathrm{L}^1\)全群中，它们的正半轨道在可能的时间反转后是可公度的。然后，卡茨内尔森准则给出了标量时间变化后的共轭性。关键的新要素是一个可公度准则，即实直线上的一个可测子集，其与平移后的对称差在单位区间上的平均测度有限，则它与空集、整条直线和两条半直线中的恰好一个是可公度的。这个准则及其应用是由一个双智能体人工智能系统自主发现的。作者独立验证了证明并准备了最终文本。
英文摘要
We prove the analogue of Belinskaya's theorem for measure-preserving flows: two free ergodic measure-preserving flows whose $\mathrm{L}^1$ full groups are isomorphic as abstract groups are conjugate up to a scalar time change. This answers a question posed by François Le Maître and the author. We show that whenever two free ergodic flows generate the same orbit equivalence relation and one is contained in the other's $\mathrm{L}^1$ full group, their positive half-orbits are commensurate after possibly reversing time. Katznelson's criterion then yields conjugacy after a scalar time change. The key new ingredient is a commensuration criterion asserting that a measurable subset of the real line whose symmetric differences with its translates have finite average measure over the unit interval is commensurate with exactly one of the empty set, the whole line, and the two half-lines. This criterion and its application were discovered autonomously by a two-agent AI system. The author independently verified the proofs and prepared the final text.
