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

Published electronically:
April 16, 2004

MathSciNet review:
2095626

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Abstract | References | Similar Articles | Additional Information

Abstract: Furstenberg showed that if two topological systems and are disjoint, then one of them, say , is minimal. When is nontrivial, we prove that must have dense recurrent points, and there are countably many maximal transitive subsystems of such that their union is dense and each of them is disjoint from . Showing that a weakly mixing system with dense periodic points is in , the collection of all systems disjoint from any minimal system, Furstenberg asked the question to characterize the systems in . We show that a weakly mixing system with dense regular minimal points is in , and each system in has dense minimal points and it is weakly mixing if it is transitive. Transitive systems in and having no periodic points are constructed. Moreover, we show that there is a distal system in .

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.

**[A]**Joseph Auslander,*Minimal flows and their extensions*, North-Holland Mathematics Studies, vol. 153, North-Holland Publishing Co., Amsterdam, 1988. Notas de Matemática [Mathematical Notes], 122. MR**956049****[AG]**Ethan Akin and Eli Glasner,*Residual properties and almost equicontinuity*, J. Anal. Math.**84**(2001), 243–286. MR**1849204**, 10.1007/BF02788112**[B]**John Banks,*Regular periodic decompositions for topologically transitive maps*, Ergodic Theory Dynam. Systems**17**(1997), no. 3, 505–529. MR**1452178**, 10.1017/S0143385797069885**[BHM]**F. Blanchard, B. Host, and A. Maass,*Topological complexity*, Ergodic Theory Dynam. Systems**20**(2000), no. 3, 641–662. MR**1764920**, 10.1017/S0143385700000341**[DY]**Tomasz Downarowicz and Xiangdong Ye,*When every point is either transitive or periodic*, Colloq. Math.**93**(2002), no. 1, 137–150. MR**1930259**, 10.4064/cm93-1-9**[F]**Harry Furstenberg,*Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation*, Math. Systems Theory**1**(1967), 1–49. MR**0213508****[GH1]**W. H. Gottschalk and G. A. Hedlund,*A characterization of the Morse minimal set*, Proc. Amer. Math. Soc.**15**(1964), 70–74. MR**0158386**, 10.1090/S0002-9939-1964-0158386-X**[GH2]**Walter Helbig Gottschalk and Gustav Arnold Hedlund,*Topological dynamics*, American Mathematical Society Colloquium Publications, Vol. 36, American Mathematical Society, Providence, R. I., 1955. MR**0074810****[HY]**Wen Huang and Xiangdong Ye,*An explicit scattering, non-weakly mixing example and weak disjointness*, Nonlinearity**15**(2002), no. 3, 849–862. MR**1901110**, 10.1088/0951-7715/15/3/320**[K]**J. Kelly, General Topology, Graduate Texts in Mathematics, 27, 1955.**[M]**Douglas C. McMahon,*Relativized weak disjointness and relatively invariant measures*, Trans. Amer. Math. Soc.**236**(1978), 225–237. MR**0467704**, 10.1090/S0002-9947-1978-0467704-9**[P]**K. E. Petersen,*Disjointness and weak mixing of minimal sets*, Proc. Amer. Math. Soc.**24**(1970), 278–280. MR**0250283**, 10.1090/S0002-9939-1970-0250283-7**[SY]**S. Shao and X. Ye,*-mixing and weak disjointness*, Topology and its Application,**135**(2004), 231-247.**[Wa]**Peter Walters,*An introduction to ergodic theory*, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR**648108****[We]**Benjamin Weiss,*A survey of generic dynamics*, Descriptive set theory and dynamical systems (Marseille-Luminy, 1996), London Math. Soc. Lecture Note Ser., vol. 277, Cambridge Univ. Press, Cambridge, 2000, pp. 273–291. MR**1774430**

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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