Absolutely extremal points in minimal flows. (English) Zbl 0591.58024

Let (T,X) be a metric flow (T is a self homeomorphism of the compact metric space X) and (T’,Q) an affine flow (i.e. Q is a compact convex subset of a locally convex linear space and T’ an affine homeomorphism). The author says a map \(\psi\) : (T,X)\(\to (T',Q)\) is an affine embedding if \(\psi\) is continuous, one to one, equivariant, and \(\overline{co}(\psi (X))=Q\). He calls a point \(x_ 0\in X\) absolutely extremal if for every embedding \(\psi\) : \(X\to Q\), \(\psi (x_ 0)\) is an extreme point of Q. (T,X) is an absolutely extremal flow if every point of X is absolutely extremal.
The author proves that distal points of a metric minimal flow (i.e. every orbit is dense) are absolutely extremal. This result generalizes a theorem of I. Namioka [Math. Z. 184, 259-269 (1983; Zbl 0505.46006)] which asserts that a distal minimal flow is absolutely extremal.
He gives an example of a weakly mixing minimal flow all of whose points are absolutely extremal and asks if there is a minimal flow no point of which is absolutely extremal. This problem has been solved by D. Maon and the author in ”On absolutely extremal points” [Compos. Math. 59, 51-56 (1986)].
Reviewer: E.Outerelo


37C10 Dynamics induced by flows and semiflows
54H20 Topological dynamics (MSC2010)
Full Text: Numdam EuDML


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