On the multiple recurrence properties for disjoint systems

Michihiro Hirayama; Dong Han Kim; Younghwan Son

doi:10.1007/s11856-021-2271-5

On the multiple recurrence properties for disjoint systems

Michihiro Hirayama
, Dong Han Kim
, Younghwan Son

Department of Mathematics Education

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

We consider a mutually disjoint family of measure preserving transformations T₁, …, T_k on a probability space (Xℬμ). We obtain the multiple recurrence property of T₁, …, T_k 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, …, T_k are uniformly jointly ergodic if each T_i is ergodic.

Original language	English
Pages (from-to)	405-431
Number of pages	27
Journal	Israel Journal of Mathematics
Volume	247
Issue number	1
DOIs	https://doi.org/10.1007/s11856-021-2271-5
State	Published - Apr 2022

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title = "On the multiple recurrence properties for disjoint systems",

abstract = "We consider a mutually disjoint family of measure preserving transformations T1, …, Tk on a probability space (Xℬμ). We obtain the multiple recurrence property of T1, …, Tk and this result is utilized to derive multiple recurrence of Poincar{\'e} 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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N2 - We consider a mutually disjoint family of measure preserving transformations T1, …, Tk on a probability space (Xℬμ). 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.

AB - We consider a mutually disjoint family of measure preserving transformations T1, …, Tk on a probability space (Xℬμ). 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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On the multiple recurrence properties for disjoint systems

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