Chaos for induced hyperspace maps. (English) Zbl 1071.37012

Banks, John

Chaos Solitons Fractals 25, No. 3, 681-685 (2005).

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.

MSC:

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)

Keywords:

sensitive dependence on initial conditions; induced maps; hyperspaces; periodic pints; topological transitivity; weak topological mixing

Citations:

Zbl 1098.37008

References:

[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