Dynamical systems disjoint from any minimal system

Abstract:

Furstenberg showed that if two topological systems $(X,T)$ and $(Y,S)$ are disjoint, then one of them, say $(Y,S)$, is minimal. When $(Y,S)$ is nontrivial, we prove that $(X,T)$ must have dense recurrent points, and there are countably many maximal transitive subsystems of $(X,T)$ such that their union is dense and each of them is disjoint from $(Y,S)$. Showing that a weakly mixing system with dense periodic points is in ${\mathcal {M}}^{\perp }$, the collection of all systems disjoint from any minimal system, Furstenberg asked the question to characterize the systems in ${\mathcal {M}}^{\perp }$. We show that a weakly mixing system with dense regular minimal points is in ${\mathcal {M}}^{\perp }$, and each system in ${\mathcal {M}}^{\perp }$ has dense minimal points and it is weakly mixing if it is transitive. Transitive systems in ${\mathcal {M}}^{\perp }$ and having no periodic points are constructed. Moreover, we show that there is a distal system in ${\mathcal {M}}^{\perp }$. Recently, Weiss showed that a system is weakly disjoint from all weakly mixing systems iff it is topologically ergodic. We construct an example which is weakly disjoint from all topologically ergodic systems and is not weakly mixing.