×

zbMATH — the first resource for mathematics

Sensitivity of set-valued discrete systems. (English) Zbl 1194.37034
Let \(X\) be a compact metric space and \(K(X)\) be the set of all nonempty compact subsets of \(X\) equipped with the Hausdorff metric. In this paper, examples are given that sensivity of a surjective continuous map \(f:X\to X\) does not imply sensivity of the induced continuous self-mapping \(\bar f:K(X)\to K(X)\). Moreover, it is shown that if \(f\) is a surjective continuous interval map, then \(\bar f\) is sensitive if and only if \(f\) has this property.

MSC:
37B99 Topological dynamics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ruelle, D.; Takens, F., On the nature of turbulence, Comm. math. phys., 20, 167-192, (1971) · Zbl 0223.76041
[2] Auslander, J.; Yorke, J., Interval maps, factors of maps and chaos, Tohoku math. J., 32, 177-188, (1980) · Zbl 0448.54040
[3] Devaney, R., Chaotic dynamical systems, (1989), Addison-Wesley Reading, MA
[4] Bauer, W.; Sigmund, K., Topological dynamics of transformations induced on the space of probability measures, Monatsh. math., 79, 81-92, (1975) · Zbl 0314.54042
[5] Fedeli, A., On chaotic set-valued discrete dynamical systems, Chaos solitons fractals, 23, 1381-1384, (2005) · Zbl 1079.37021
[6] Liao, G.F.; Wang, L.D.; Zhang, Y.C., Transitivity, mixing and chaos for a class of set valued mappings, Sci. China ser. A math., 49, 1-8, (2006) · Zbl 1193.37023
[7] Peris, A., Set-valued discrete chaos, Chaos solitons fractals, 26, 19-23, (2005) · Zbl 1079.37024
[8] Banks, J., Chaos for induced hyperspace maps, Chaos solitons fractals, 25, 681-685, (2005) · Zbl 1071.37012
[9] Román-Flores, H., A note on transitivity in set-valued discrete systems, Chaos solitons fractals, 17, 99-104, (2003) · Zbl 1098.37008
[10] Kwietniak, D.; Oprocha, P., Topological entropy and chaos for maps induced on hyperspaces, Chaos solitons fractals, 33, 76-86, (2007) · Zbl 1152.37306
[11] Gu, R.B., Kato’s chaos in set-valued discrete systems, Chaos solitons fractals, 31, 765-771, (2007) · Zbl 1140.37305
[12] Román-Flores, H.; Chalco-Cano, Y., Robinson’s chaos in set-valued discrete systems, Chaos solitons fractals, 25, 33-42, (2005) · Zbl 1071.37013
[13] Wang, Y.; Wei, G.; Campbell, W.H., Sensitive dependence on initial conditions between dynamical systems and their induced hyperspace dynamical systems, Topol. appl., 156, 803-811, (2009) · Zbl 1172.37006
[14] García-Guirao, J.L.; Kwietniak, D.; Lampart, M.; Oprocha, P.; Peris, A., Chaos on hyperspaces, Nonlinear anal., 71, 1-8, (2009) · Zbl 1175.37024
[15] Robinson, C., Dynamical systems, (1995), CRC Press Rockville, MD
[16] Barge, M.; Martin, J., Chaos, periodicity and snakelike continua, Trans. amer. math. soc.., 289, 355-365, (1985) · Zbl 0559.58014
[17] Blokh, A.M., On sensitive mappings of the interval translation, Russ. math. surv., 37, 203-204, (1982) · Zbl 0511.28013
[18] Furstenberg, H., Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. system th., 1, 1-55, (1967) · Zbl 0146.28502
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.