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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
##### Keywords:
set-valued mapping; sensitivity; path-connected space
Full Text:
##### 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) · Zbl 0695.58002 [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) · Zbl 0853.58001 [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