The capturing operation in topological dynamics (Q2850661)

By a flow \((X,T)\) is understood a continuous action of a topological group \(T\) on a compact Hausdorff space \(X\). It is called minimal if for every \(x\in X\) the closure \(\overline{xT}\) of its orbit \(xT\) equals \(X\). Two points \(x\) and \(x'\) of a flow \((X,T)\) are said to be proximal if there is a net \(\{t_i\}\) in \(T\) such that \((xt_i, x't_i)\to (y,y)\) for some \(y\in X\). If \(x\) and \(x'\) are not proximal they are said to be distal. The flow \((X,T)\) is distal if \(P=\Delta\), where \(\Delta\) is the diagonal in \(X\times X\) and \(P\) is the proximity relation. For the group \(T\) there exists a unique universal minimal flow \((M,T)\) defined by the property that every minimal flow \((X,T)\) is a factor of \((M,T)\). Let \(G\) be the automorphism group of \((M,T)\). If \(\alpha\in G\), the graph of \(\alpha\) is \(\Gamma_\alpha= \{(m,\alpha(m))\mid m\in M\}\) and if \(A\subset G\), then \(\Gamma_A= \bigcup\{\Gamma_\alpha\mid \alpha\in A\}\). If \((X,T)\) is a minimal flow and \(\pi:M\to X\) a homomorphism, then the Ellis group of \((X,T)\) is the subgroup of \(G\) defined by \({\mathfrak G}(X)= \{\alpha\in G\mid\pi\alpha= \pi\}\). There is a compact \(T_1\) topology on \(G\) such that \({\mathfrak G}(X)\) is closed. If \((X,T)\) is a flow and \(K\subset X\), the capturing set of \(K\) is the set \(C(K)= \{x\in X\mid\overline{xT}\cap K\neq\emptyset\}\) and the strong capturing set \(C^*(K)= \{x\in X\mid\overline{xT}\subset K\}\). If the capturing operation \(C\) is applied to the product flow \((X\times X,T)\), then it becomes a relation of \(X\). A closed subgroup \(A\) of \(G\) is said to be distal, if there is a distal minimal flow \((X,T)\) with \({\mathfrak G}(X)= A\). Among other results obtained the author proves that if \(A\) is a closed subgroup of \(G\), then the following statements are equivalent:NEWLINENEWLINE (i) \(A\) is a distal group.NEWLINENEWLINE (ii) \(C(\Gamma_A)\) is closed.NEWLINENEWLINE (iii) \(C^*(\Gamma_A)\) is closed and \(C(\Gamma_A)\) is an equivalence relation.