On systems disjoint from all minimal systems
Wen Huang, Song Shao, Hui Xu, Xiangdong Ye
TL;DR
The paper characterizes when a topological dynamical system is disjoint from all minimal systems, yielding intrinsic criteria that sharpen previous ergodic analogues. It proves that a system $X$ is disjoint from all minimal systems iff it admits a dense assembly of minimal subsets each disjoint from $X$, and, in the semi-simple case, is characterized by a residual relation on orbit closures; it also develops a topological decomposition framework via almost one-to-one extensions and maximal equicontinuous factors, including for distal systems. A central technical contribution is the countability of pairwise disjoint quasifactors for a minimal system and the derived residual criteria that connect disjointness to hyperspace dynamics and regionally proximal relations. The results unify and extend prior observations on disjointness from minimal and distal systems, offering intrinsic, verifiable conditions with implications for transitive hyperspace dynamics and ergodic-type decompositions in the topological setting.
Abstract
Recently, Górska, Lemańczyk, and de la Rue characterized the class of automorphisms disjoint from all ergodic automorphisms. Inspired by their work, we provide several characterizations of systems that are disjoint from all minimal systems. For a topological dynamical system $(X,T)$, it is disjoint from all minimal systems if and only if there exist minimal subsets $(M_i)_{i\in\mathbb{N}}$ of $X$ whose union is dense in $X$ and each of them is disjoint from $X$ (we also provide a measure-theoretical analogy of the result). For a semi-simple system $(X,T)$, it is disjoint from all minimal systems if and only if there exists a dense $G_δ$ set $Ω$ in $X \times X$ such that for every pair $(x_1,x_2) \in Ω$, the subsystems $\overline{\mathcal{O}}(x_1,T)$ and $\overline{\mathcal{O}}(x_2,T)$ are disjoint. Furthermore, for a general system a characterization similar to the ergodic case is obtained.