The Dynamic System Of Transitive Extreme Value Distributional Chaos On The Whole Space

As it known to all, dynamical system is the important part of nonlinearscience. Topological dynamical system and ergodic theory has madeconsiderable achievements in the early1930s. Since then, the dynamicalsystem has been developing along two parallel lines: one is the discoveryand research the simplicity, stability and predictability of system; Anotheris to explain the complexity, instability and chaos of system.Speaking of chaos, chaotic behavior is decided in the dynamicalsystem which seemed a randomly movement. Its essence is the sensitivityto initial conditions of long-term behavior of the system. However, we donot have a clear concept of chaos in a long time until1975, Li and Yorkeput forward mathematical definition for the first time in Period threeimplies chaos. After that, chaos has a more and more important status inthe research of the dynamical system. With the deepening of the researchon Li-Yorke chaos, the experts gave different definitions from differentangles of chaos. Among them, distributional chaos is a very importantconcept which is introduced by Schweizer and SmItal according to thedistribution of asymptote of the distance between two tracks. The firstproblem in distributional chaos researchs is to point out the relationshipbetween all kinds of chaos, and to known Whether it equivalent topositive topological entropy and topological mixing or not. Based on this, scholars have studied on distributional chaos in the recent years. In orderto have a better understanding of the distributional chaos, this papersummarizes the distributional chaos of the existing research results inrecent years.Distributional chaos is similar to Li-Yorke chaos but more complicated.As both distributional chaos and Li-Yorke chaos is defined by thescrambled set, the study on the size and property of scrambled set is acrucial part in the research of chaos. In this paper, through theconstruction of a group of finite sequence{E n}n1, we construct anon-compact dynamical system, in which each point’s orbit is dense inthe whole space. So the whole space is both invariant extremedistributional scrambled set and a transitive distributional scrambled set.In addition, a extreme value distributional chaos on the whole space isconstructed.