Chaos for induced hyperspace maps

By J. Banks. 2005

149 citations recorded by the source corpus.

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. Roman-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 Roman-Flores.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/12d6287fe2160083066333e6a6a2cf8d7f2374c3

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In Applied Mathematics

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Citations in the corpus

Inducing sensitivity on hyperspaces (2010, background): “there have been many attempts to study relations between the dynamical properties of the map f and that of the induced map f [3,7,9–12].”
Recent development of chaos theory in topological dynamics (2015, background): “1 ([11, 13]) Let (X,T ) be a dynamical system.”
Topological entropy and chaos for maps induced on hyperspaces (2007, unknown): “In recent years this question has attracted many researchers and their results have been published by this (see [1–8]) and other journals (see [9,10]).”