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

Abstract

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The properties of topological dynamical systems ( X , T ) which are disjoint from all minimal systems of zero entropy, 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 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  ℤ + , which contains a subset of  ℤ + arising from a positive entropy minimal system, but does not contain any subset of  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.

How to cite

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Huang, 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

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  12. [12] W. Huang & X. Ye – « Generic eigenvalues, generic factors and weak disjointness », Preprint. Zbl1293.37012MR2931915
  13. [13] —, « Dynamical systems disjoint from any minimal system », Trans. Amer. Math. Soc.357 (2005), p. 669–694. Zbl1072.37011MR2095626
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