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Ultrafilter-limit points in metric dynamical systems. (English) Zbl 1199.54194

If \(X\) is a compact space, \(f\) a continuous self-map of \(X\) and \(p\) a free ultrafilter on the set of natural numbers, the authors define \(f^p\) to be the pointwise limit of the iterates \(f^n\), \(n=1,2,\dots \), along the ultrafilter \(p\). The paper contains characterizations of continuity of \(f^p\) (at a point or on the whole space \(X\)), some results on the relationship of \(f^p\) and \(f^q\) for two distinct ultrafilters \(p,q\), a study of particular properties of \(f^p\) for a \(P\)-point \(p\) and some results relating this kind of limits to actions on metrizable semigroups.

MSC:

54H20 Topological dynamics (MSC2010)
54D80 Special constructions of topological spaces (spaces of ultrafilters, etc.)
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