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Chaos for induced hyperspace maps. (English) Zbl 1071.37012

Summary: For $\left(X,d\right)$ be a metric space, $f:X\to X$ a continuous map and $\left(𝒦\left(X\right),H\right)$ the space of nonempty compact subsets of $X$ with the Hausdorff metric, one may study the dynamical properties of the induced map

$\overline{f}:𝒦\left(X\right)\to 𝒦\left(X\right):A↦f\left(A\right)·$

H. Román-Flores [Chaos Solitons Fractals 17, 99–104 (2003; Zbl 1098.37008)] has shown that if $f$ is topologically transitive then so is $f$, but that the reverse implication does not hold. This paper shows that the topological transitivity of $\overline{f}$ is in fact equivalent to weak topological mixing on the part of $f$. This is proved in the more general context of an induced map on some suitable hyperspace $ℋ$ of $X$ with the Vietoris topology which agrees with the topology of the Hausdorff metric in the case discussed by Roman-Flores.

##### MSC:
 37B99 Topological dynamics 37A25 Ergodicity, mixing, rates of mixing 37B05 Transformations and group actions with special properties 54H20 Topological dynamics