On Scrambled Sets And Its Related Problems

In the present paper, we study mainly some properties of scrambled sets, the car-dinalities of scrambled sets related to subshift of finite type via Furstenberg families.In Chapter 1, firstly, the origin and main contents of the topological dynamicalsystem are presented. Secondly, we mainly introduce the purpose and contend of thepaper.In Chapter 2, we introduce the basic notions and properties of Furstenberg fam-ilies and subshift of finite type of symbolic space on topological dynamical system.In Chapter 3, some properties with respect to scrambled sets are studied viaFurstenberg families. For a positive integer k, the properties P(k) and Q(k) ofFurstenberg families are defined. It is shown that for any s∈[0, 1], the Fursten-berg family M(s) has the properties P(k) and Q(k), where M(s) denotes the familyof all infinite subsets of Z+ whose upper density is not less than s. Furthermore, weprove that for any positive integer k, S is an (M(s),M(t))-scrambled set of (X,f) ifand only if S is an (M(s),M(t))-scrambled set of (X,fk), where s,t∈[0, 1].In Chapter 4, we discuss cardinalities of maximal scrambled sets with respect tosubshift of finite type. We show that for any subshift of finite type (ΣA,σA) deter-mined by 0, 1?matrix A which each element of l row and each element of l column is1, where l∈{0,1,···,N ? 1}, if it is topologically transitive and not minimal, thenevery maximal scrambled sets of the subshift has cardinality c; Moreover, we point outthat for any subshift of finite type (ΣA,σA) which have two fixed points and each ofmaximal (F1, F2)-scrambled sets has cardinality c, if F1 and F2 are two compatibletoΣA×ΣA and kB ? F1,kB ? F2.