Homeomorphisms of hyperspaces7/5/2023 ![]() ![]() Furthermore, we discuss the sensitivity for the systems onhyperspaces induced by M-mappings. According to this result, theM-systems are classified completely: if (X,f) is an M-system, then f satisfies oneof the following four properties exactly, (a) f is equicontinuous (b) f is minimaland weakly mixing (c) f is minimal but neither equicontinuous nor weakly mixing (d) f is non-minimal. Moreprecisely,In the preface, we retrospect the evolution of dynamical system, set-valuedanalysis and operator theory.In the first chapter, some preliminary knowledge in dynamical system, hyper-space and operator theory is reviewed, which will be used in this paper.In chapter 2, a result belonging to Glasner and Weiss is referred: an M-systemis either sensitive or minimal and equicontinuous. In addition, Read constructed a bounded linearoperator on l1 without nontrivial invariant subspace, then it is interesting to studythe topological conjugacy between two operators, since we may obtain, if possibly,a topological conjugacy to get a bounded linear operator on Hilbert space withoutnontrivial invariant subspace from Read’s operator.Our basic objective in this paper is to study the sensitivity for the systemson hyperspaces induced by M-mappings and some dynamical aspects for boundedlinear operators such as sensitivity, orbit, topological conjugacy and so on. ![]() ![]() Topo-logical conjugacy is a weaker relation than similarity, and consequently it is morecomprehensive without linearity. Feldman researched the sensitivity for operators, andcooperated with Bourdon to discuss the properties of orbits for operators. ![]() Grosse-Erdmann, Costakis and Sambarino studied chaos and strongly mixing for boundedlinear operators respectively. gave a su?cient condition to transitivity - Hy-percyclicity Criterion, which is also a su?cient condition to weakly mixing Ansariproved the equivalence between transitivity and totally transitivity, etc. There have been many results about transitive operators: for a bounded linear operator, Kitai et al. In fact, the notion”hypercyclic”coincides with”transitivity”in dynamical system. On the other hand, an important task for operator theory is to study the con-struction of operator, which could be researched from the point of dynamical system.For the well-known open problem about invariant subspace, the hypercyclic opera-tor is interesting. So it is necessary to study the hyperspace dynamical system(K(X),f) induced by the dynamical system (X,f), where K(X) is the family of allof the nonempty compact sets of X. There are natural joints to linking them, and we emphasis the mutual in?uenceof classic thought, notions and conclusions in these theories in order to acceleratethem developing together.One hand, in many fields and problems such as biological species, demography,numerical simulation and attractors, etc., it is not enough to know only how thepoints of X move, one has to know how the subsets of X move, especially for thecompact subsets. In this thesis, we link topological dynamical system with set-valuedanalysis and operator theory, those are often said set-valued system and linear sys-tem. A direction for the development of modern mathematics is to put variousbranches across. ![]()
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