# Topological disjointness from entropy zero systems

Wen Huang; Kyewon Koh Park; Xiangdong Ye

Bulletin de la Société Mathématique de France (2007)

- Volume: 135, Issue: 2, page 259-282
- ISSN: 0037-9484

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topHuang, Wen, Koh Park, Kyewon, and Ye, Xiangdong. "Topological disjointness from entropy zero systems." Bulletin de la Société Mathématique de France 135.2 (2007): 259-282. <http://eudml.org/doc/272311>.

@article{Huang2007,

abstract = {The properties of topological dynamical systems $(X,T)$ which are disjoint from all minimal systems of zero entropy, $\mathcal \{M\}_0$, are investigated. Unlike the measurable case, it is known that topological $K$-systems make up a proper subset of the systems which are disjoint from $\mathcal \{M\}_0$. We show that $(X,T)$ has an invariant measure with full support, and if in addition $(X,T)$ is transitive, then $(X,T)$ is weakly mixing. A transitive diagonal system with only one minimal point is constructed. As a consequence, there exists a thickly syndetic subset of $\{\mathbb \{Z\}\}_+$, which contains a subset of $\{\mathbb \{Z\}\}_+$ arising from a positive entropy minimal system, but does not contain any subset of $\{\mathbb \{Z\}\}_+$ arising from a zero entropy minimal system. Moreover we study the properties of topological dynamical systems $(X,T)$ which are disjoint from larger classes of zero entropy systems.},

author = {Huang, Wen, Koh Park, Kyewon, Ye, Xiangdong},

journal = {Bulletin de la Société Mathématique de France},

keywords = {disjointness; minimality; entropy; density},

language = {eng},

number = {2},

pages = {259-282},

publisher = {Société mathématique de France},

title = {Topological disjointness from entropy zero systems},

url = {http://eudml.org/doc/272311},

volume = {135},

year = {2007},

}

TY - JOUR

AU - Huang, Wen

AU - Koh Park, Kyewon

AU - Ye, Xiangdong

TI - Topological disjointness from entropy zero systems

JO - Bulletin de la Société Mathématique de France

PY - 2007

PB - Société mathématique de France

VL - 135

IS - 2

SP - 259

EP - 282

AB - The properties of topological dynamical systems $(X,T)$ which are disjoint from all minimal systems of zero entropy, $\mathcal {M}_0$, are investigated. Unlike the measurable case, it is known that topological $K$-systems make up a proper subset of the systems which are disjoint from $\mathcal {M}_0$. We show that $(X,T)$ has an invariant measure with full support, and if in addition $(X,T)$ is transitive, then $(X,T)$ is weakly mixing. A transitive diagonal system with only one minimal point is constructed. As a consequence, there exists a thickly syndetic subset of ${\mathbb {Z}}_+$, which contains a subset of ${\mathbb {Z}}_+$ arising from a positive entropy minimal system, but does not contain any subset of ${\mathbb {Z}}_+$ arising from a zero entropy minimal system. Moreover we study the properties of topological dynamical systems $(X,T)$ which are disjoint from larger classes of zero entropy systems.

LA - eng

KW - disjointness; minimality; entropy; density

UR - http://eudml.org/doc/272311

ER -

## References

top- [1] F. Blanchard – « Fully positive topological entropy and topological mixing », in Symbolic dynamics and its applications (New Haven, CT, 1991), Contemp. Math., vol. 135, Amer. Math. Soc., 1992, p. 95–105. Zbl0783.54033MR1185082
- [2] —, « A disjointness theorem involving topological entropy », Bull. Soc. Math. France121 (1993), p. 465–478. Zbl0814.54027MR1254749
- [3] F. Blanchard, B. Host & A. Maass – « Topological complexity », Ergodic Theory Dynam. Systems20 (2000), p. 641–662. Zbl0962.37003MR1764920
- [4] F. Blanchard & Y. Lacroix – « Zero entropy factors of topological flows », Proc. Amer. Math. Soc.119 (1993), p. 985–992. Zbl0787.54040MR1155593
- [5] H. Furstenberg – « Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation », Math. Systems Theory1 (1967), p. 1–49. Zbl0146.28502MR213508
- [6] —, Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, 1981, M. B. Porter Lectures. Zbl0459.28023MR603625
- [7] E. Glasner – « A simple characterization of the set of $\mu $-entropy pairs and applications », Israel J. Math.102 (1997), p. 13–27. Zbl0909.54035MR1489099
- [8] E. Glasner & B. Weiss – « Quasi-factors of zero-entropy systems », J. Amer. Math. Soc.8 (1995), p. 665–686. Zbl0846.28009MR1270579
- [9] —, « Locally equicontinuous dynamical systems », Colloq. Math. 84/85 (2000), p. 345–361, Dedicated to the memory of Anzelm Iwanik. Zbl0976.54044MR1784216
- [10] W. H. He & Z. L. Zhou – « A topologically mixing system whose measure center is a singleton », Acta Math. Sinica (Chin. Ser.) 45 (2002), p. 929–934 (Chinese). Zbl1010.37003MR1941883
- [11] W. Huang, S. Shao & X. Ye – « Mixing via sequence entropy », Comtemp. Math.385 (2005), p. 101–122. Zbl1103.37002MR2180232
- [12] W. Huang & X. Ye – « Generic eigenvalues, generic factors and weak disjointness », Preprint. Zbl1293.37012MR2931915
- [13] —, « Dynamical systems disjoint from any minimal system », Trans. Amer. Math. Soc.357 (2005), p. 669–694. Zbl1072.37011MR2095626
- [14] W. Parry – « Zero entropy of distal and related transformations », in Topological Dynamics (Symposium, Colorado State Univ., Ft. Collins, Colo., 1967), Benjamin, New York, 1968, p. 383–389. Zbl0193.51602MR234443
- [15] B. Song & X. Ye – « A minimal completely positive, non uniformly positive entropy example », to appear in J. Difference Equations and Applications. Zbl1154.37010MR2484420
- [16] B. Weiss – « Topological transitivity and ergodic measures », Math. Systems Theory5 (1971), p. 71–75. Zbl0212.40103MR296928

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