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Proximity and distality via Furstenberg families. (English) Zbl 1099.37006

Summary: Proximity, distality and recurrence are studied via Furstenberg families. A new proof of some classical results on the conditions when a proximal relation is an equivalence one is given. Moreover, for a family \(\mathcal F\), \(\mathcal F\)-almost distality and \(\mathcal F\)-semi-distality are defined and characterized. As an application a new characterization of PI-flows is obtained.

MSC:

37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37B20 Notions of recurrence and recurrent behavior in topological dynamical systems
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