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A note on transitivity in set-valued discrete systems. (English) Zbl 1098.37008

Summary: Let \((X,d)\) be a metric space and \(f:X \to X\) is a continuous function. If we consider the space \((\mathcal K(X),H)\) of all non-empty compact subsets of \(X\) endowed with the Hausdorff metric induced by \(d\) and \(\bar f : \mathcal K(X) \to \mathcal K(X)\), \(\bar f(A) = \{f(a)/a \in A\}\), then the aim of this work is to show that \(\bar f\) transitive implies \(f\) transitive. Also, we give an example showing that \(f\) transitive does not implies \(\bar f\) transitive.

MSC:

37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
54H20 Topological dynamics (MSC2010)
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