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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Family independence for topological and measurable dynamics


Authors: Wen Huang, Hanfeng Li and Xiangdong Ye
Journal: Trans. Amer. Math. Soc. 364 (2012), 5209-5242
MSC (2010): Primary 37B40, 37A35, 37B10, 37A05
Published electronically: May 2, 2012
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Abstract: For a family $ \mathcal {F}$ (a collection of subsets of $ \mathbb{Z}_+$), the notion of $ \mathcal {F}$-independence is defined both for topological dynamics (t.d.s.) and measurable dynamics (m.d.s.). It is shown that there is no non-trivial {syndetic}-independent m.d.s.; an m.d.s. is {positive-density}-independent if and only if it has completely positive entropy; and an m.d.s. is weakly mixing if and only if it is {IP}-independent. For a t.d.s. it is proved that there is no non-trivial minimal {syndetic}-independent system; a t.d.s. is weakly mixing if and only if it is {IP}-independent.

Moreover, a non-trivial proximal topological K system is constructed, and a topological proof of the fact that minimal topological K implies strong mixing is presented.


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Additional Information

Wen Huang
Affiliation: Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China
Email: wenh@mail.ustc.edu.cn

Hanfeng Li
Affiliation: Department of Mathematics, Chongqing University, Chongqing 401331, People’s Republic of China – and – Department of Mathematics, SUNY at Buffalo, Buffalo, New York 14260-2900
Email: hfli@math.buffalo.edu

Xiangdong Ye
Affiliation: Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China
Email: yexd@ustc.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05493-6
PII: S 0002-9947(2012)05493-6
Keywords: Independence, weak mixing, minimal, K
Received by editor(s): December 14, 2009
Received by editor(s) in revised form: August 25, 2010
Published electronically: May 2, 2012
Additional Notes: The first author was partially supported by the NNSF of China (10911120388), the Fok Ying Tung Education Foundation, FANEDD (Grant 200520) and the Fundamental Research Funds for the Central Universities
The second author was partially supported by NSF grant DMS-0701414
The first and third authors were partially supported by grants from NNSF of China (10531010, 11071231) and 973 Project (2006CB805903)
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.