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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

##### MSC:
 37C10 Dynamics induced by flows and semiflows 54H20 Topological dynamics (MSC2010)
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##### References:
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