2023 Fiscal Year Final Research Report
Research in quantitative aspects of multiple recurrence property of probability measure preserving transformations
| Project/Area Number |
19K03558
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 12020:Mathematical analysis-related
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| Research Institution | University of Tsukuba |
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Principal Investigator |
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| Project Period (FY) |
2019-04-01 – 2024-03-31
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| Keywords | 多重再帰定理 / Khintchine型多重再帰定理 / Furstenberg独立性 / 極小自己結合 / Mobius直交性 |
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| Outline of Final Research Achievements |
Recurrence is considered as one of the most fundamental and important property in ergodic theory, and much research has been conducted from various perspectives. This property is extended from the case of a single transformation to multiple transformations. Regarding the quantitative aspects of multiple recurrence property, we conducted complementary research on the general theory and an investigation of specific examples. For families of probability measure-preserving transformations that are disjoint in the sense of Furstenberg, we established the mean convergence of multiple ergodic averages and the multiple recurrence result of the Khintchine type. We also showed that certain specific measure-preserving transformation has minimal self joining.
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| Free Research Field |
エルゴード理論
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| Academic Significance and Societal Importance of the Research Achievements |
再帰性はエルゴード理論における基本的かつ重要な性質の一つであり，様々な観点から多くの研究がなされている．しかしながら，保測変換族に対する多重同時再帰性の定量的側面については，まだ明らかにされていないことが多い．この点について，本研究ではFurstenbergの意味で独立な保測変換族が呈する同時再帰時間のなす集合が，自然数全体において相対稠密であることを明らかにした（Khintchine型の多重再帰定理）．変換族が可換な場合には先行する結果があったが，本研究成果は可換とは限らない変換族にも応用をもつ点で学術的意義があると思われる．
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