Discrete Dynamical Systems For Some Hyperspaces

It is well known that the main task to investigate the dynamical system(X,f) is to clear how the points of X move. Nevertheless, in many fields andproblems such as biological species, demography, numerical simulation andattractors, etc., it is not enough to know only how the points of X move, onehas to know how the subsets of X move, especially for the compact subsets. Soit is necessary to study the hyperspace dynamical system (K(X),f) induced bythe dynamical system (X,f), where K(X) is the family of all of the nonemptycompact sets of X .In 2003, Roman-Flores first compare the transitivity between the discretedynamical system (X,f) and its induced hyperspace system (K(X),f). Hisresults show that the transitivity of the induced hyperspace system (K(X),f)implies that of the base system (X,f) and that the transitivity on both sys-tems are not equivalent. Moreover, he ask a fundamental question about thechaos: Can the chaos on the base systems imply the chaos on the hyperspacessystems? and conversely? After that, the research on the relations betweenthe individual chaos and collective chaos springs up.This question of Roman-Flores attract the interest of many scholar in avery short time. In 2004, Fedeli proved that the periodically dense on the basesystem implies the same property on its induced hyperspace system. Later,Peris, Gu Rongbao, Liao Gongfu and the author of this paper investigate themixing, conclude that the transitivity, the weakly mixing on the hyperspacesystem is equivalent to the weakly mixing on the base system, the stronglymixing on both systems are equivalent. J. Banks consider a family of hyper-spaces where each one of them can be dense at the power set space of thebase space, he proved that the above conclusion is also right for each suchhyperspace system. When the base space is graph, Zhang Gengrong provedthat Devaney chaos of the hyperspace system strictly implies that of the basesystem, also, even with the coarser topology, We-topology, the inverse is notcorrect. Gu Rongbao proved that Kato chaos of the hyperspace system strictlyimplies that of the base system , in the sense of We-topology, Kato chaos onboth systems are equivalent. In 2005, Roman-Flores concluded that Robin-son chaos on the hyperspace system strictly implies Robinson chaos on basesystem.However, for those obviously important properties which re?ect the com-plexity of the dynamical systems, there is no results, especially on Li-Yorkechaos, Distribution chaos, positive topological entropy and Devaney chaos.That is to say: in the framework of Roman-Flores' question, can Li-Yorkechaos of the hyperspace system is equivalent to that of the base system? whatabout the Distribution chaos, the positive topological entropy? what relationsbetween the hyperspace system and the base system on Devaney chaos?This paper resolved the above problems and other problems about thestability, the properties related with transitivity, etc.. The following are theresults in detail:On the stability aspects.The base system is equicontinuous if and only if its induced hyperspacesystem is equicontinuous. For isometry, the results is same. The productsystem of a sensitive system and an isometric system is also sensitive . As anexample, the hyperspace system of the cylinder system is sensitive , where thecylinder system is the product system of the classical tent map system on theinterval and the isometric rotation system on the circle.On the properties related with transitivity.The topological exactness, weakly mixing and strongly mixing are equiv-alent between the two systems, but it is not equivalent on extreme scatter-ing, strong scattering, scattering, weak scattering, total transitivity and n-transitivity, respectively. However, on the hyperspace system, transitivity,n-transitivity, total transitivity, weak scattering, scattering, strong scattering,extreme scattering, weakly mixing are equivalent each other. The hyperspacesystem induced by the dynamical system which is total transitive and period-ically dense is Devaney chaos.For the special situation on the interval, more equivalent results can beobtained. In particular, 2-transitivity on the base system is equivalent to thestrongly mixing on its induced hyperspace system.On the aspects of the complexity.The emphasis lays on the research of some canonical properties which re-?ect complexity of the dynamical system, such as positive topological entropy,Li-Yorke chaos, Distribution chaos and Devaney chaos.For Li-Yorke chaos, positive topological entropy and Distribution chaos,these properties on base system can imply those of its induced hyperspacesystem, respectively. A system with zero topological entropy and no Li-Yorkepairs is constructed, whose induced hyperspace system is Distribution chaosand has positive topological entropy. This fact suggests that Li-Yorke chaos,positive topological entropy and Distribution chaos on the hyperspace systemcan not imply those of the base system, respectively. The result also show thatPena and Lopez's conclusion about the equivalence about positive topologicalentropy is wrong and the equivalence declared by Lampart on the Li-Yorkechaos is not correct on the dynamical system and its induced hyperspacesystem.For Devaney chaos, the strongly mixing minimal system has at most oneperiodic point and it is also the fixed point of the hyperspace system. Thisresult suggests that it is impossible to consider the equivalence of Devaneychaos on such systems. A dynamical system with topological strong mixingand only one periodic point is constructed, whose induced hyperspace systemis periodically dense. This construction suggests that the periodically densityof the hyperspace system can not imply that of the base system, even withthe condition of topological strong mixing. Moreover, Devaney chaos on thehyperspace system can not imply Devaney chaos on the base system.In conclusion, the above results give a satisfied answer for those problemsproposed by Roman-Flores, Fedeli, J. Banks and Liao Gongfu, etc..