Some Dynamical Properties For The Systems On Hyperspaces Induced By M-mappings And Linear Operators

A direction for the development of modern mathematics is to put variousbranches across. 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. 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. 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 .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. In fact, the notion"hypercyclic"coincides with"transitivity"in dynamical system. There have been many results about transitive operators: for abounded linear operator, Kitai et al. 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.. Grosse-Erdmann, Costakis and Sambarino studied chaos and strongly mixing for boundedlinear operators respectively. Feldman researched the sensitivity for operators, andcooperated with Bourdon to discuss the properties of orbits for operators. Topo-logical conjugacy is a weaker relation than similarity, and consequently it is morecomprehensive without linearity. 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. 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. 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. Furthermore, we discuss the sensitivity for the systems onhyperspaces induced by M-mappings. As well-known, if f satisfies the property of (a), then f is equicontinuous; if f satisfies the property of (b), then f is weaklymixing and hence is sensitive; Liu Heng constructed an example to show that asystem on hyperspace induced by a M-mapping satisfying the property of (c) couldhave an equicontinuous point.We prove that a system on hyperspace induced by a non-minimal M-mapping issensitive; The Cartesian product of a weakly mixing minimal system and an additivemachine is minimal but neither equicontinuous nor weakly mixing, and the system onhyperspace induced by it is sensitive, this implies a system on hyperspace inducedby an M-mapping satisfying the property of (c) can be sensitive. Therefore, theprevious conclusions extend the result of Glasner and Weiss, and perfect the theoryof sensitivity for the systems on hyperspaces induced by M-mappings.In chapter 3, some dynamical properties for bounded linear operators are con-sidered. Sensitivity is preserved by the topological conjugacy between two boundedlinear operators. As a byproduct, if two bounded linear operators are topologi-cally conjugate, then there exists a topological conjugacy h between them such thath(0) = 0. As viewed from topological dynamical system, we reprove the equivalentdescriptions of transitivity and mixing for backward unilateral weighted shift oper-ators in an elementary way. We prove that a continuous mapping from a metricspace to another is locally bounded. Consequently, it is preserved, by the topolog-ical conjugacy between two bounded linear operators, to have a bounded discreteorbit. Here an orbit is called discrete, if itsω-limit set is empty. Specially, if abackward unilateral weighted shift operator is Devaney chaotic, then the closure ofeach bounded orbit is compact and hence it can't be discrete. Moreover, there ex-ists a backward unilateral weighted shift operator having a bounded discrete orbit,we construct such an example as follows: S is a backward unilateral weighted shiftoperator on l2 with the weighted sequence {ωn}n∞=1, In chapter 4, the topologically conjugate classification is considered. First ofall, we construct a family of homeomorphisms from lp onto itself and validate themto be. Let S = {sn}n∞=1 be a sequence of positive numbers satisfying sn≥1 for eachn≥1. The mapping hSp is defined on lp as follows, for every x = (x1,x2,...)∈lp,whereπn is the projection to the n?th coordinate. Then via choosing the sequenceof parameters S properly, and considering the dynamical properties preserved bytopological conjugacy between two bounded linear operators, we prove the followingresults:If two bounded positive diagonal operators satisfying that the infimums of theirdiagonal sequences respectively are both strictly more than 1 (or the infimums areboth strictly more than 0 and the supremums are both strictly less than 1), thenthey are topologically conjugate. And we give an example to show that the strictnessfor the sign of inequalities in previous condition is necessary;We give a complete topologically conjugate classification for constant-weightedbackward unilateral shift operators: two constant-weighted backward unilateral shiftoperators are topologically conjugate if and only if the modules of the two weightsare synchronously more than 1, or equivalent to 1 or less than 1. Moreover, list somebackward unilateral weighted shift operators which are not topologically conjugateeach other.