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Proximality in minimal transformation groups. (English) Zbl 0202.23302
This paper presents an action of the integers on a compact metric space such that in the resulting minimal topological transformation group the proximal relation is an equivalence relation but is not closed. It had been conjectured that such an example did not exist, and many partial results in that direction have appeared in the literature. The example given here also seems to be the first known example of an inverse limit and homomorphic image of expansive transformation groups which is not itself expansive, but this seems unrelated to the example’s other properties.

MSC:
54H20 Topological dynamics (MSC2010)
54H15 Transformation groups and semigroups (topological aspects)
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
54E05 Proximity structures and generalizations
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