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| Ergodic Behaviour of Nonconventional Ergodic Averages for Commuting Transformations |
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| KeyWord:Commuting transformation convergence almost everywhere ergodic behaviour time average space average |
| Author Name | Affiliation | | Xia Pan | College of Liberal Arts and Sciences, National University of Defense Technology, Changsha, Hunan 410073, China | | Zuohuan Zheng | Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
College of Mathematics and Statistics, Hainan Normal University, Haikou, Hainan 571158, China | | Zhe Zhou | Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China |
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| Abstract: |
| Based on T. Tao’s celebrated result on the norm convergence of multiple ergodic averages for commuting transformations, we find that there is a subsequence which converges almost everywhere. Meanwhile, we obtain the ergodic behaviour of diagonal measures, which indicates the time average equals the space average. According to the classification of transformations, we also give several different results. Additionally, on the torus Td with special rotation, we prove the pointwise convergence in Td, and get a result for ergodic behaviour. |
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