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Chaos for induced hyperspace maps. (English) Zbl 1071.37012

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

f ¯:𝒦(X)𝒦(X):Af(A)·

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 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 of X with the Vietoris topology which agrees with the topology of the Hausdorff metric in the case discussed by Roman-Flores.

37B99Topological dynamics
37A25Ergodicity, mixing, rates of mixing
37B05Transformations and group actions with special properties
54H20Topological dynamics