The Chaotic Properties Of Infinite-dimensional Linear System

This thesis is devoted to describing the chaotic properties of the infinite-dimensionallinear systems. It consists of four chapters. In chapter one, we introduce some clas-sical results about chaos phenomena in topological dynamical system and infinite-dimensional linear system. In chapter two, we recall some basic concepts, includingseveral definitions of chaos, such as sensitive dependence on initial conditions, Devaneychaos and Li Yorke chaos. In chapter three, we study the relationships between severalchaotic phenomena in infinite-dimensional linear system, and calculate the topologicalentropy of diagonal operators. The concrete results are as follows:(1) A topological transitive linear system with a periodic point has an uncount-ably dimensional linear subspace which is a scrambled set;(2) If a linear system has positive topological entropy, then the system has sen-sitive dependence on initial conditions;(3) For a diagonal operator T=diag(λ1,...λn,...), we have:h(K, T)=ilog λi, for any compact subset K of X,If limnâ†'∞n log n log λn=0, then h(X, T)=ilog λi,If K=[0, a1]×[0, a2]×..., and a21+a22+...<∞, then h(K, T)=ilog λi.