Family independence for topological and measurable dynamics (Q2841400)

Let \((X, T)\) be a topological dynamical system (t.d.s.) in which \(X\) is a compact metrizable space and \(T\) is a surjective continuous map from \(X\) to itself. For a tuple \(A = (A_1,\dots,A_k)\) of subsets of \(X\), a subset \(F\subset\mathbb{Z}_+\) is said to be an \textit{independence set} for \(A\) if for any nonempty finite subset \(J \subset F\), NEWLINE\[NEWLINE \bigcap_{j\in J}T^{-j}A_{s(j)}\neq \emptyset NEWLINE\]NEWLINE for any \(s\in\{1, \dots , k\}^J\). Denote the collection of all independence sets for \(A\) by \(\mathrm{Ind}A\). For a family \(\mathcal{F}\) (a collection of subsets of \(\mathbb{Z}_+\)), a t.d.s. \((X, T)\) is said to be \(\mathcal{F}\)-\textit{independent}, if for each \(k\in\mathbb{N}\), for each tuple of nonempty open subsets \(U_1,\dots, U_k\), \(\mathrm{Ind}(U_1,\dots, U_k) \cap \mathcal{F}\neq \emptyset\). Let \((X,\mathcal{B},\mu,T)\) be a measurable dynamical system (m.d.s.) where \((X,\mathcal{B},\mu)\) is a Lebesgue space and \(T:X\longrightarrow X\) is measurable and \(\mu\)-measure-preserving. A notion of \(\mathcal{F}\)-\textit{independence} for \((X,\mathcal{B},\mu,T)\) can be introduced in a similar way.NEWLINENEWLINEIn this paper, the authors investigate the following question: for a given family \(\mathcal{F}\) which dynamical property is equivalent to \(\mathcal{F}\)-independence? It is shown that there is no non-trivial syndetic-independent m.d.s.; a m.d.s. is positive-density-independent if and only if it has completely positive entropy; and a m.d.s. is weakly mixing if and only if it is IP-independent. For a t.d.s. it is proved that there is no non-trivial minimal syndetic-independent system; a t.d.s. is weakly mixing if and only if it is IP-independent. Moreover, a non-trivial proximal topological \(K\) system is constructed, and a topological proof of the fact that minimal topological \(K\) implies strong mixing is presented.