Sensitivity And Attractor In Discrete Dynamical Systems

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, Rom′an-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 systemsare not equivalent.This question of Rom′an-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 Ma Xianfeng paper investigate the mix-ing, conclude that the transitivity, the weakly mixing on the hyperspace systemis equivalent to the weakly mixing on the base system, the strongly mixing onboth systems are equivalent. J. Banks consider a family of hyperspaces whereeach one of them can be dense at the power set space of the base space, heproved that the above conclusion is also right for each such hyperspace system.When the base space is graph, Zhang Gengrong proved that Devaney chaosof the hyperspace system strictly implies that of the base system, also, evenwith the coarser topology, We-topology, the inverse is not correct. Gu Rong-bao proved that Kato chaos of the hyperspace system strictly implies that of the base system , in the sense of We-topology, Kato chaos on both systemsare equivalent. In 2005, Rom′an-Flores concluded that Robinson chaos on thehyperspace system strictly implies Robinson chaos on base system.However, for sensitivity which is on the core station in topology dynami-cal system and some properties which re?ect the complexity of the dynamicalsystems and associate with sensitivity, there is no results. That is to say: inthe framework of Rom′an-Flores'question, can sensitivity(Li-Yorke sensitivity)of the base system imply sensitivity(Li-Yorke sensitivity) of the hyperspacesystem? What condition we need such that Li-Yorke chaos of the base sys-tem imply Li-Yorke chaos of the hyperspace system? Can Kato chaos of thehyperspace system is equivalent to that of the base system?On the other hand, an important task in dynamical system is the investi-gation on attractor. It's well known that every adic system is minimal, uniqelyergodic and has zero topological entropy. So if a system has a adic attractorthen most points have simple dynamical behavior, we say it is total stable insome sense. While sensitivity which is total unstable and positive topologyentropy are important characters of complex system. It is nature to ask: ifor not there exists a map which has both an adic attractor and positive topo-logical entropy; and what relations between the existence of an adic attractorand sensitivity?Our basic objective in this paper is to study the relations between sen-sitivity and correlative properties of basic system and those of hyperspacesystem, the relation between the existence of an adic attractor and positivetopological entropy and the relation between the existence of an adic attractorand sensitivity. We emphasis the investigation on complex properties with to-tal unstable characters, such as sensitivity, Li-Yorke sensitivity, Kato's chaosetc., and relation between the existence of an adic attractor which has totalstable character and sensitivity. More precisely,In chapter 1, some preliminary knowledge in topologically dynamical sys-tem, which will be used in this paper, are reviewed.In chapter 2, the relation between sensitivity of the basic system and thatof the hyperspace system and a kind of generalized sensitivity are studied.A system (S,F) on path-connected space is constructed,which is sensi- tive, but (K(S),F) is not sensitive , where S = {(r,2πθ) : r∈I,θ∈Cα} be asubset in polar coordinate system with metricρdefined byThe continuous map F : S→S is defined by F(r,2πθ) = (f(r),2πDα(θ))(where (Cα,Dα) is a concrete Denjoy ). The construct of counter examplemainly recur to the Denjoy homeomorphism (Cα,Dα) , Dαis sensitive, buton hyperspace system, Cαis a stable point of Dα. On interval, it is shownthat sensitivity of the basic system is equivalent to that of the hyperspacesystem.Finally, from the idea of Prof. Xiong ( [96]), the definition of totallymaximum sensitive (TMS for short) is introduced: Let (X,d) be a compactmetric space containing at least two points and f : X→X be continuous,We say f is totally maximum sensitive (TMS for short), ifλn = rn for eachn∈N, and prove that f is weakly mixing i? it is TMS.In chapter 3 the relation between the existence of an adic attractor andpositive topological entropy and the relation between the existence of an adicattractor and sensitivity are studied.For a interval map with an adic attractor, by simply classifying attractor,we can get the status of topological entropy on the system generally. we prove:the condition that n is not a power of 2 is su?cient but not neccesary for themap to have positive topological entropy. A counter example is given to showthat the inverse version of above proposition doesn't hold, i.e., there existsa continuous self-map of [0, 1] with adic attractor, such that it has positivetopological entropy while it's attractor is not n-adic if the n is not a power of2. And a class of solutions of the functional equation f3(λx) =λf(x), whichwill be proved to have 3-adic attractor, are investigated. Finally, it is shownthat if (I,f) has an adic attractor, then f is not sensitive.In chapter 4, the relations between some properties correlative with sen-sitivity of basic system and those of hyperspace system are studied.It is shown that if the continuous map of the compact metric space(X,d) without isolated points into itself is transitive but non-minimal, then(K(X),f) is Li-Yorke chaotic. Furthermore, we prove that if (X,f) is non-minimal M?system, and V , V are minimal subsets contained in X with d(V,V ) > 4s, where s is a positive numver, then there exists a Cantor setC such that S = {V∪{x} : x∈C} is a s?scrambled set of f. Also, wegive counter example to prove that there exist system (X,f) such that (X,f)is Li-Yorke sensitive( Kato chaotic), but (K(X),f) is not sensitive( Katochaotic).These results develop the theory of set-valued discrete dynamical system.