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Topological weak specification and distributional chaos on noncompact spaces. (English) Zbl 1497.37016

Summary: In this paper, we relate the topological definitions of specification property and distributional chaos defined for uniformly continuous self-maps on noncompact, nonmetrizable spaces. We prove that a uniformly continuous surjective self-map acting on a uniformly locally compact Hausdorff uniform space with topologically weak specification property and a pair of distal points is topologically distributionally chaotic of type 1. This extends the result due to P. Oprocha and M. Štefánková [Proc. Am. Math. Soc. 136, No. 11, 3931–3940 (2008; Zbl 1159.37004)]. As a consequence, we get that uniformly continuous surjective self-map on a uniformly locally compact totally bounded Hausdorff uniform space with topological shadowing, topological mixing, and a distal pair is topologically distributionally chaotic of type 1.

MSC:

37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37B02 Dynamics in general topological spaces
37B40 Topological entropy

Citations:

Zbl 1159.37004
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References:

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