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 | 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]**J. Auslander, Minimal Flows and Their Extensions, North-Holland Mathematics Studies, 153. North-Holland Publishing Co., Amsterdam, 1988. MR**89m:54050****[AG]**E. Akin and E. Glasner,*Residual properties and almost equicontinuity*, J. d'Anal. Math.,**84**(2001), 243-286. MR**2002f:37020****[B]**J. Banks,*Regular periodic decomposition for topologically transitive maps*, Ergod. Th. and Dynam. Sys.,**17**(1997), 505-529. MR**98d:54074****[BHM]**F. Blanchard, B. Host and A. Maass,*Topological complexity*, Ergod. Th. and Dynam. Sys.,**20**(2000), 641-662. MR**2002b:37019****[DY]**T. Downarowicz and X. Ye,*When every point is either transitive or periodic*, Colloq. Math.,**93**(2002), 137-150. MR**2003g:37015****[F]**H. Furstenberg,*Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation*, Math. Systems Theory,**1**(1967), 1-49. MR**35:4369****[GH1]**W. Gottschalk and G. Hedlund,*A characterization of the Morse minimal set*, Proc. Amer. Math. Soc.,**15**(1964), 70-74. MR**28:1609****[GH2]**W. Gottschalk and G. Hedlund, Topological Dynamics, Amer. Math. Soc. Colloq., Vol. XXXVI, 1955. MR**17:650e****[HY]**W. Huang and X. Ye,*An explicit scattering, nonweakly mixing example and weak disjointness*, Nonlinearity,**15**(2002), 849-862. MR**2003b:37016****[K]**J. Kelly, General Topology, Graduate Texts in Mathematics, 27, 1955.**[M]**D. McMahon,*Relativized weak disjointness and relatively invariant measures*, Trans. Amer. Math. Soc.,**236**(1978), 225-237. MR**57:7557****[P]**K. Petersen,*Disjointness and weak mixing of minimal sets*, Proc. Amer. Math. Soc.,**24**(1970), 278-280. MR**40:3522****[SY]**S. Shao and X. Ye,*-mixing and weak disjointness*, Topology and its Application,**135**(2004), 231-247.**[Wa]**P. Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York, Berlin, 1982. MR**84e:28017****[We]**B. Weiss, A survey of generic dynamics, LMS Lecture Note Series, 277, Cambridge University Press, 2000, 273-291. MR**2001j:37008**

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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