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Dynamical systems disjoint from any minimal system


Authors: Wen Huang and Xiangdong Ye
Journal: Trans. Amer. Math. Soc. 357 (2005), 669-694
MSC (2000): Primary 54H20; Secondary 58K15
DOI: https://doi.org/10.1090/S0002-9947-04-03540-8
Published electronically: April 16, 2004
MathSciNet review: 2095626
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Abstract: Furstenberg showed that if two topological systems $(X,T)$ and $(Y,S)$ are disjoint, then one of them, say $(Y,S)$, is minimal. When $(Y,S)$ is nontrivial, we prove that $(X,T)$ must have dense recurrent points, and there are countably many maximal transitive subsystems of $(X,T)$ such that their union is dense and each of them is disjoint from $(Y,S)$. Showing that a weakly mixing system with dense periodic points is in ${\mathcal{M}}^{\perp }$, the collection of all systems disjoint from any minimal system, Furstenberg asked the question to characterize the systems in ${\mathcal{M}}^{\perp }$. We show that a weakly mixing system with dense regular minimal points is in ${\mathcal{M}}^{\perp }$, and each system in ${\mathcal{M}}^{\perp }$ has dense minimal points and it is weakly mixing if it is transitive. Transitive systems in ${\mathcal{M}}^{\perp }$ and having no periodic points are constructed. Moreover, we show that there is a distal system in ${\mathcal{M}}^{\perp }$.

Recently, Weiss showed that a system is weakly disjoint from all weakly mixing systems iff it is topologically ergodic. We construct an example which is weakly disjoint from all topologically ergodic systems and is not weakly mixing.


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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

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: https://doi.org/10.1090/S0002-9947-04-03540-8
Keywords: Disjoint, weakly disjoint, minimal, scattering, weakly mixing
Received by editor(s): November 1, 2002
Received by editor(s) in revised form: July 15, 2003
Published electronically: April 16, 2004
Additional Notes: The research of the second author was supported by the 973 project
Article copyright: © Copyright 2004 American Mathematical Society

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