Topological disjointness from entropy zero systems

Huang, Wen; Koh Park, Kyewon; Ye, Xiangdong

Bulletin de la Société Mathématique de France, Tome 135 (2007) no. 2, pp. 259-282.

Nous étudions les propriétés des systèmes topologiques dynamiques $(X,T)$ 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 $(X,T)$ 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 ${\mathbb{Z}}_{+}$, qui contient un sous-ensemble de ${\mathbb{Z}}_{+}$, provenant d’un système minimal d’entropie positive, mais qui ne contienne aucun sous-ensemble de ${\mathbb{Z}}_{+}$ 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.

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 ${\mathbb{Z}}_{+}$, 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.

Classification : 54H20
Keywords: disjointness, minimality, entropy, density
Mot clés : disjonction, minimalité, entropie, densité

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