Reply To The Point Of The Existence Of Dynamical Systems On Locally Compact

The core issue of the dynamical system is the progressive nature or topology of the orbit of point. We also know that only those points with recurrence are important, while the recurrent point is a definition used to describe the progres-sive nature of the orbit of point. In other words, a set of recurrent points can indirectly characterize the complexity of the dynamical system. At present, as for compact dynamical system, we can know from Birkhoff's theorem of recur-rent point that there exist recurrent points in the compact dynamical system. However, strict restrictions are imposed for dynamic system when compactness is claimed for its base space so that the most important dynamical system topol-ogy, such as those with Rn as their base space etc. cannot be considered as compact dynamical systems. In this paper, we put aside the compactness of the base space of the dynamical space and focus on locally compact dynamical system whose scale is much larger. In addition, we give different situations to talk about whether recurrent points exist or not respectively.The specific contents are as follows:In the first chapter, I'd like to introduce the historical development of dy-namical system and some definitions for basic items in this area. Then the importance of the study of recovery in dynamical system is discussed and the current results of this study are released. And I hope the study of locally compact dynamical system can develop in this direction. Finally, the research purposes and main results are simply introduced.In the second chapter, we put aside the compactness of the base space in the dynamical system and give the definition of locally compact dynamical system. In the preliminaries, the definition of continuous map f converge at infinity and the fact that continuous map f can be expended as a necessary and sufficient condition are introduced respectively.Before the main results are released, four examples are given to elaborate the special locally compact dynamical system with R as its base space. Due to these examples, we can know whether the given continuous map can be extended, the existence of the recurrent point in locally compact dynamical system is uncertain. so, it is essential we classify the question into more detailed.In the last chapter, From what we have learned in the above examples and the preliminaries, I'd like to group those questions into several types to discuss and give their results. At last in this paper, we especially consider the locally compact dynamical system (Rn,f).Finally, we sum up the main results, innovations of this paper and some researches which are to be studied in the future.