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

Nous étudions les propriétés des systèmes topologiques dynamiques (X,T) qui sont disjoints de tout système minimal d’entropie nulle 0 . Contrairement au cas mesurable, il est connu que les K-systèmes topologiques constituent un sous-ensemble propre des systèmes disjoints de 0 . Nous montrons que (X,T) a une mesure invariante à support plein, et que si, de plus, (X,T) 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  + , qui contient un sous-ensemble de  + , provenant d’un système minimal d’entropie positive, mais qui ne contienne aucun sous-ensemble de  + provenant d’un système minimal d’entropie nulle. D’autre part, nous étudions les propriétés des systèmes topologiques dynamiques (X,T) 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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