Research On Chaoticity And Shadowing Properties Of Dynamical Systems

This thesis studies the chaoticity and shadowing property with an average error and mainly completes the following four tasks:1. Consider dynamical properties under iteration, inverse limit, hyperspace and g-fuzzification and obtain the following results:(1) A non-autonomous discrete system which converges uniformly is (?)-chaotic if and only if its n-th iterate system is (?)-chaotic for any n∈N, where & denotes one of the eight properties:Li-Yorke chaos, dense chaos, dense δ-chaos, generic chaos, generic δ-chaos, Li-Yorke sensitivity, sensitivity, spatiotemporal chaos, DC1-chaos, DC2-chaos. Besides, it is proved that a product system is multi-sensitive if and only if some factor system is multi-sensitive. This is a positive answer to an open question posed by Li and Zhou (2013) in Turkish Journal of Mathematics.(2) Firstly, it is proved that a dynamical system is weakly mixing (resp., topologi-cally mixing) if and only if its hyperspace system is topologically transitive (resp., topo-logically mixing) if and only if its Zadeh’s extension system is topologically transitive (resp., topologically mixing), which is equivalent to the weakly mixing property of its g-fuzzification. Secondly, a sufficient condition is obtained to ensure that for every con-tinuous self-map defined on X, its g-fuzzification system is not topologically transitive. Thirdly, it is proved that if the g-fuzzification system is sensitively dependent, then the hyperspace system is sensitively dependent. Finally, using some examples it is shown that there exists a sensitive dynamical system whose g-fuzzification system is not sensi-tively dependent for any g∈Dm(I). These results give negative answers to some open questions about sensitivity and weakly mixing property posed by Kupka (2014) in Infor-mation Sciences.2. For the full shift (∑2,σ) on two symbols, an invariant distributionally ε-scrambled set for all 0<ε< diamE2 is constructed. Besides, for the following system f:Σ2×S1(?)(x,t)â†'(σ(x),Rrx1(t))∈∑2 x S1, (?)x=x1x2…E E2,(?)t∈S1, where S1={e2εiθ:0≤0<1}(?)C, it is proved that (∑2 x S1,f) has an uncountable distributionally β-scrambled set for any 0<β< diam∑2 xS11. This is a positive answer to an open question posed by Wang et al. (2003) in Annales Polonici Mathematici.3. Study the chaoticity of linear systems. Firstly, it is proved that for a continuous linear operator defined on a Banach space, Li-Yorke chaos, distributionally chaos in a sequence, Li-Yorke sensitivity, and spatiotemporal-chaos are all equivalent, and they are all strictly stronger than sensitivity. Then, the chaoticity of the weighted shift operator Bw defined on the Kothe sequence space λP(A) is studied, obtaining a class of equivalent conditions which give Li-Yorke chaos for Bw. Besides, it is proved that if there exist a pair x,y∈λP(A) and 8> 0 such that liminfnâ†'∞(1/n)|{0<j<n:d(Bwj(x),Bwj(y))<δ}|< 1, then there exists ε> 0 such that Bw admits an uncountable invariant DC2-ε-scrambled subset for some ε> 0 and Bw admits an invariant DC2-scrambled linear manifold. More-over, a sufficient condition is obtained to ensure that Bw admits an invariant distribution-ally ε-scrambled subset for any 0< ε< diamλp(A). As a corollary, it follows that the principal measure of an annihilation operator a=(?)(x+d/dx) of the unforced quantum harmonic oscillator is equal to 1. This gives an exact answer to a open question posed by Oprocha (2006) in Journal of Physics A:Mathematical and General.4. Investigate the shadowing property with an average error in tracing a dynamical system. Firstly, it is proved that Mα-shadowing property and Mα-shadowing property are both preserved under iterations; implying that if dynamical system f|A has the Mα-shadowing property for some a ∈[0,1) on A, then so does f on X, where A C X is a closed invariant set containing the measure center of a compact dynamical system (X,f). Moreover, a dynamical system (X,f) has the almost specification property if and only if (Y,f|Y) has the almost specification property, where Y is the measure center. So, the almost specification property implies the asymptotic average shadowing property. This gives a partial positive answer to an open question posed by Kulczycki, Kwietniak, and Oprocha (2014) in Fundamenta Mathematicae. Next, using the obtained results and Mα-shadowing property, it is shown that for general cases (the map needs not be onto), the following statements hold: almost specification property(?)AASP(?)WASP (?)AASP(?)Mα-shadowing property,(?)α∈(0,1] (?)d-shadowing property+d-shadowing property.This result improves the main results about the relations of various shadowing properties of Kulczycki, Kwietniak, and Oprocha. Finally, the relations among shadowing proper- ties, transitive property and sensitivity are discussed. In particular, it is proved that there exists no nontrivial equicontinuous surjective dynamical system with the d-shadowing property or d-shadowing property.