Semigroup compactifications by generalized distal functions and a fixed point theorem (Q1175433)

For a flow \((S,X)\), let \(E\) denote the enveloping semigroup, \(E=S^ - \subset X^ X\). \((S,X)\) is called distal if \(E\) is a group. When \(S\) is a semigroup without identity, one can generalize the concept of distality in the following way. For each \(n\in\mathbb{N}\), set \(E^ n=\left\{ \prod^ n_ 1 g_ i\mid g_ i\in E\right\}^ -\subset E\), and set \(E^{\infty}=\bigcap_ 1^{\infty}E^ n\). Then \((S,X)\) is called \(n\)- distal (\(\infty\)-distal) if \(\xi(x)=\xi(x')\) for some \(\xi\in E\) implies \(\zeta(x)=\zeta(x')\) for all \(\zeta\in E^ n\) (\(\zeta\in E^{\infty})\). The author studies these ideas, and also introduces the related notions of \(n\)- and \(\infty\)-distal functions, for which he determines the relevant universal mapping properties and fixed point theorems (the latter involving invariant means in the usual way).