Chaos for induced hyperspace maps

John Banks

Chaos, Solitons & Fractals, 2005, vol. 25, issue 3, 681-685

Abstract: For (X,d) be a metric space, f:X→X a continuous map and (K(X),H) the space of non-empty compact subsets of X with the Hausdorff metric, one may study the dynamical properties of the induced map (∗)f¯:K(X)→K(X):A↦f(A).H. Román-Flores [A note on in set-valued discrete systems. Chaos, Solitons & Fractals 2003;17:99–104] 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 H of X with the Vietoris topology (which agrees with the topology of the Hausdorff metric in the case discussed by Román-Flores.

Date: 2005
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Citations: View citations in EconPapers (14)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:25:y:2005:i:3:p:681-685

DOI: 10.1016/j.chaos.2004.11.089

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