Research On Distributionally Chaotic Dynamics Of Infinite Dimensional Linear Systems

As a characterization of complexity of dynamical systems, chaos exists in nature widely. Chaotic phenomenon is intimately linked to nonlinearity in finite dimensional spaces. However, chaos for linear systems has been also discovered when the phase space is infinite dimensional. Infinite dimensional linear dynamics is concerned with the long time behaviors of semi-dynamical systems generated by iterates of linear operators on infinite dimensional spaces and by strongly continuous semigroups of linear operators(C0-semigroups for short). Particularly, the study of hypercyclicity and other chaotic properties of linear operators consumedly enlarges and develops the theory of linear chaos,it also suggests a new way of thinking about the essence of chaos. Meanwhile, these results are well applied to many natural systems from different disciplines, including statistical mechanics, quantum mechanics, biology, economics and engineering. Consequently, detailed research on chaotic behaviors of infinite dimensional linear systems is not only necessary but also significance in theory and application.In this dissertation, the distributionally chaotic dynamics of infinite dimensional linear systems is studied extensively. By using some methods of topological dynamical systems and related theory of operators, we investigate the principal measure, the properties of distributionally scrambled pairs, distributionally scrambled sets and distributionally n-scrambled sets for linear operators and for C0-semigroups, respectively. We also prove the property of distributional chaos and the existence of invariant distributionally scrambled linear manifolds for several concrete linear systems. The detailed framework of this dissertation is as follows.Chapter 1 gives a brief survey to the background of this dissertation, the research history and advance of chaotic dynamical systems, some related concepts and results in topological dynamics and linear chaos and the main works of this dissertation.Chapter 2 studies the generically distributionally chaotic dynamics of linear operators and C0-semigroups on Fr′echet spaces. Sufficient conditions for a C0-semigroup on a Fr′echet space to be generically distributionally chaotic are provided and applied to concrete examples. Meanwhile, distributional chaos and principal measure of product operators(product C0-semigroups, respectively) are further concerned. The obtained results are applied to construct distributionally chaotic operators, which are not generically(or densely) distributionally chaotic, or not Devaney chaotic, respectively. Another interesting finding is that there exist distributionally chaotic(but not hypercyclic) operators whose principal measure could be less than any fixed positive number.Chapter 3 considers some properties of distributionally scrambled sets for an annihilation operator of the unforced quantum harmonic oscillator, including the size, the invariant property, the diversity, the algebraic structure and topological structure. We prove the annihilation operator admits infinitely many densely invariant distributionally scrambled linear manifolds, and admits no residual distributionally scrambled set. These results provide new insight on the further study of complexity of general linear systems.Chapter 4 investigates distributional n-chaos and several properties of distributionally n-scrambled sets for infinite dimensional linear systems. We prove that an operator(or a C0-semigroup) exhibits distributional chaos if and only if it is distributionally nchaotic for any integer n 2, while this property does not hold for topological dynamical systems. We also show that the whole space could be a distributionally n-scrambled set for some linear systems. Finally, we focus on distributional n-chaos for a class of composition operators C?, showing that C?admits an uncountable distributionally scrambled set which is not distributionally 3-scrambled, and even a finite dimensional distributionally scrambled linear manifold of C?may not be distributionally 3-scrambled.Chapter 5 studies the chaotic dynamics of a class of weighted shift operators. For the case of unilateral shift, we show that this operator is distributionally chaotic with principal measure one, and we also construct densely invariant distributionally n-scrambled linear manifolds for it. Further, we turn to the bilateral weighted shift and prove the maximal distributionally chaotic property of this operator and of its inverse. We prove that there exist dense linear manifolds that are distributionally scrambled of the bilateral weighted shift, but not of its inverse; there are also dense linear manifolds that are both invariant,distributionally scrambled of the bilateral weighted shift and of its inverse. Moreover,we construct an uncountable set which is distributionally scrambled both of the bilateral weighted shift and of its inverse, but this set can not be a distritbuionally 3-scrambled set neither of the bilateral weighted shift nor of its inverse. Finally, we further prove these weighted shifts are all topologically mixing and Devaney chaotic.