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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 $(\cal K(X),H)$ of all non-empty compact subsets of $X$ endowed with the Hausdorff metric induced by $d$ and $\bar f : \cal K(X) \to \cal K(X)$, $\bar f(A) = \{f(a)/a \in A\}$, 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.

37B05Transformations and group actions with special properties
54H20Topological dynamics
Full Text: DOI
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