Chaos for induced hyperspace maps. (English) Zbl 1071.37012

Summary: For \((X, d)\) be a metric space, \(f: X\to X\) a continuous map and \(({\mathcal K}(X),H)\) the space of nonempty compact subsets of \(X\) with the Hausdorff metric, one may study the dynamical properties of the induced map \[ \overline f:{\mathcal K}(X)\to{\mathcal K}(X): A\mapsto f(A). \] 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 \(\overline f\) 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 \({\mathcal H}\) of \(X\) with the Vietoris topology which agrees with the topology of the Hausdorff metric in the case discussed by Roman-Flores.


37B99 Topological dynamics
37A25 Ergodicity, mixing, rates of mixing
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
54H20 Topological dynamics (MSC2010)


Zbl 1098.37008
Full Text: DOI Link


[1] Banks, J., Regular periodic decompositions for topologically transitive maps, Discrete Cont Dynam Sys, 5, 83-92 (1997) · Zbl 0921.54029
[2] Banks, J., Topological mapping properties defined by digraphs, Discrete Cont Dynam Sys, 5, 83-92 (1999) · Zbl 0957.54020
[3] Banks, J.; Brooks, J.; Cairns, G.; Davis, G.; Stacey, P., On Devaney’s definition of chaos, Amer Math Mon, 99, 332-334 (1992) · Zbl 0758.58019
[4] Barnsley, M., Fractals everywhere (1993), Academic Press: Academic Press London · Zbl 0691.58001
[5] Devaney, R., An introduction to chaotic dynamical systems (1989), Addison Wesley: Addison Wesley Menlo Park, CA · Zbl 0695.58002
[6] Edalat, A., Dynamical Systems, Measures and fractals via domain theory, Inform Comput, 120, 32-48 (1995) · Zbl 0834.58029
[7] Furstenberg, H., Disjointness in ergodic theory, minimal sets and a problem in diophantine approximation, Syst Theory, 1, 1-49 (1967) · Zbl 0146.28502
[8] Hutchinson, J., Fractals and self similarity, Indiana U Math J, 30, 713-747 (1981) · Zbl 0598.28011
[9] Michael, E., Topologies on spaces of subsets, Trans. Amer. Math. Soc., 71, 152-182 (1951) · Zbl 0043.37902
[10] Román-Flores, H., A note on in set-valued discrete systems, Chaos, Solitons & Fractals, 17, 99-104 (2003) · Zbl 1098.37008
[11] Silverman, S., On maps with dense orbits and the definition of chaos, Rocky Mt J Math, 22, 353-375 (1992) · Zbl 0758.58024
[12] Wicks, K., Fractals and hyperspaces. Fractals and hyperspaces, Lecture Notes In Mathematics 1492 (1991), Springer-Verlag: Springer-Verlag Berlin · Zbl 0770.54048
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.