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On the multiple recurrence properties for disjoint systems

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Abstract

We consider a mutually disjoint family of measure preserving transformations T1, …, Tk on a probability space \((X,{\cal B},\mu )\). We obtain the multiple recurrence property of T1, …, Tk and this result is utilized to derive multiple recurrence of Poincaré type in metric spaces. We also present the multiple recurrence property of Khintchine type. Further, we study multiple ergodic averages of disjoint systems and we show that T1, …, Tk are uniformly jointly ergodic if each Ti is ergodic.

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Acknowledgments

The authors wish to thank the anonymous referee for the valuable comments and suggestions. MH was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Number 19K03558. DK was supported by the National Research Foundation of Korea (NRF-2018R1A2B6001624). YS was supported by the National Research Foundation of Korea (NRF-2020R1A2C1A01005446).

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Authors and Affiliations

  1. Department of Mathematics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki, 305-8571, Japan

    Michihiro Hirayama

  2. Department of Mathematics Education, Dongguk University-Seoul, 30 Pildong-ro 1-gil, Jung-gu, Seoul, 04620, Korea

    Dong Han Kim

  3. Department of Mathematics, POSTECH, 77 Cheongam-Ro, Nam-Gu, Pohang, Gyeongbuk, 37673, Korea

    Younghwan Son

Authors
  1. Michihiro Hirayama
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  2. Dong Han Kim
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  3. Younghwan Son
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Correspondence to Dong Han Kim.

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Hirayama, M., Kim, D.H. & Son, Y. On the multiple recurrence properties for disjoint systems. Isr. J. Math. 247, 405–431 (2022). https://doi.org/10.1007/s11856-021-2271-5

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  • DOI: https://doi.org/10.1007/s11856-021-2271-5

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