On disjointness and $\mathscr{F}$-product recurrence with respect to zero entropy systems

Abstract

This paper studies disjointness and \(\mathscr{F}\)-product recurrence with respect to zero-entropy systems. First, we prove that, for a point whose orbit closure has zero topological entropy, \[ \text{distality}\rightleftharpoons \Finf\PRzero\rightleftharpoons \Fpubd\PRzero \rightleftharpoons \Fps\PRzero \rightovernotleft \Fs\PRzero. \] We also construct a minimal topological dynamical system with uniform positive entropy which is disjoint from all zero-entropy \(M\)-systems but is not disjoint from some zero-entropy \(E\)-system, i.e., \(\Ezero^{\perp}\subsetneq \Mzero^{\perp}\), giving an affirmative answer to a question of \cite[W.~Huang, K.~K. Park and X.~Ye, Topological disjointness from entropy zero systems, Bull. Soc. Math. France \textbf{135} (2007), 259--282]. Moreover, the same construction shows that there exists an \(\Fps\PRzero\) point which is not \(\Fpubd\PRzero\), i.e., $\Fps\PRzero$ $\nRightarrow \Fpubd\PRzero$, giving a negative answer to a question of \cite[P.~Oprocha and G.~H. Zhang, On weak product recurrence and synchronization of return times, Adv. Math. \textbf{244} (2013), 395--412].