Topological aspects of dynamics of pairs, tuples and sets (Q2874870)

The authors study dynamical systems \((X,f)\) where \(f\) is a continuous surjection of a compact metric space \((X,d)\) onto itself. If \((X,f)\) and \((Y,g)\) are two such systems and \(\sigma:X\to Y\) is a continuous surjection with \(g\circ \sigma= \sigma\circ f\), then \((Y,g)\) is called a factor of \((X,f)\) and \((X,f)\) is called an extension of \((Y,g)\). The authors recall the recent result that each system \((X,f)\) has a maximal factor with zero entropy. Two points \(x\), \(y\) of \((X,f)\) are called proximal if \(\liminf_{n\to\infty}\{d(f^n(x), f^n(y)\}= 0\). If for each \(\varepsilon> 0\) the set \(\{n\in\mathbb{N}: d(f^n(x), f^n(y)<\varepsilon\}\) is syndetic, then \(x\) and \(y\) are called syndetically proximal. Let Prox\((f)\) and Syn\,Prox\((f)\) denote the set of proximal and syndetically proximal pairs, respectively. A positive orbit of a point \(x\) in \((X,f)\) is denoted by \(\text{Orb}^+(x,f)\) and its closure by \(\overline{\text{Orb}^+(x,f)}\). A point \(x\in X\) is called minimal if \([\overline{\text{Orb}^+(x,f)},f]\) is a minimal dynamical system. The authors cite the result due to Auslander that for every \(x\in X\) there is a minimal point \(y\) in \(\overline{\text{Orb}^+(x,f)}\) such that \(x\) and \(y\) are proximal. Among other results the following theorem is obtained: For any dynamical system \((X,f)\) the relation Syn\,Prox\((f)\) is an equivalence relation and the following conditions are equivalent:NEWLINENEWLINE (1) Prox\((f)\) is an equivalence relation.NEWLINENEWLINE (2) \(\text{Prox}(f)= \text{Syn\,Prox}(f)\).NEWLINENEWLINE (3) The closure of the orbit of any point \((x,y)\in X^2\) in the system \((X^2,f^2)\) contains exactly one minimal set.NEWLINENEWLINEFor the entire collection see [Zbl 1282.54001].