Topological disjointness from entropy zero systems
Bulletin de la Société Mathématique de France, Volume 135 (2007) no. 2, p. 259-282

The properties of topological dynamical systems $\left(X,T\right)$ 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 $\left(X,T\right)$ has an invariant measure with full support, and if in addition $\left(X,T\right)$ is transitive, then $\left(X,T\right)$ 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 $\left(X,T\right)$ which are disjoint from larger classes of zero entropy systems.

Nous étudions les propriétés des systèmes topologiques dynamiques $\left(X,T\right)$ qui sont disjoints de tout système minimal d’entropie nulle ${ℳ}_{0}$. Contrairement au cas mesurable, il est connu que les $K$-systèmes topologiques constituent un sous-ensemble propre des systèmes disjoints de ${ℳ}_{0}$. Nous montrons que $\left(X,T\right)$ a une mesure invariante à support plein, et que si, de plus, $\left(X,T\right)$ est transitif alors il est faiblement mélangeant. Nous construisons un système diagonal transitif avec un seul point minimal. Par conséquent, il existe un sous-ensemble grassement syndétique de ${ℤ}_{+}$, qui contient un sous-ensemble de ${ℤ}_{+}$, provenant d’un système minimal d’entropie positive, mais qui ne contienne aucun sous-ensemble de ${ℤ}_{+}$ provenant d’un système minimal d’entropie nulle. D’autre part, nous étudions les propriétés des systèmes topologiques dynamiques $\left(X,T\right)$ qui sont disjoints de classes plus larges de systèmes à entropie nulle.

DOI : https://doi.org/10.24033/bsmf.2534
Classification:  54H20
Keywords: disjointness, minimality, entropy, density
@article{BSMF_2007__135_2_259_0,
author = {Huang, Wen and Koh Park, Kyewon and Ye, Xiangdong},
title = {Topological disjointness from entropy zero systems},
journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
publisher = {Soci\'et\'e math\'ematique de France},
volume = {135},
number = {2},
year = {2007},
pages = {259-282},
doi = {10.24033/bsmf.2534},
zbl = {1157.54015},
mrnumber = {2430193},
language = {en},
url = {http://www.numdam.org/item/BSMF_2007__135_2_259_0}
}

Huang, Wen; Koh Park, Kyewon; Ye, Xiangdong. Topological disjointness from entropy zero systems. Bulletin de la Société Mathématique de France, Volume 135 (2007) no. 2, pp. 259-282. doi : 10.24033/bsmf.2534. http://www.numdam.org/item/BSMF_2007__135_2_259_0/

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