About Rotation Number And Expanding Map Of Homeomorphism Onto Itself For Torus In Topological Space

In the 1880s,the research of dynamic system had been originated with a French m- athematicians H.Poincare,who published continuously papers Differential Equation To DetermineCurve.It established differential equation qualitative theory and it was also c- alled differential equation geometry theory.This theory didn't through differential equat- ion demonstration solution ,but studies directly the geometry and the topological proper- ty.In the early 20th century ,the research of dynamic system obtained the amazing prog- ress because of the cantact of practical issues such as nonlinear vibration.Specially G.D. Birkhoff abstracted topological dynamical system from the dynamic system which was definedby the classical differential equation.They enabled the discipline to further obtai- n the development theoretically and made dynamic system become a richest achieveme- nt mathematics branch in the 20th ,and obtained the substantial progress.The dynamic s- ystem has become a important component in non-linear science in modern mainstream discipline.At the same time,dynamic system has not only been studied in non-linear dis- cipline,but also is a powerful tool in learning the complexity in non-linear discipline.Th- e theory has been infiltrated widely in many important domains.In the process of the dynamic system in compact low-dimension manifold,the cov- ering and inflation mapping are very useful topological tools.It is very convenient to study the property of self-mapping of circumference by using the covering space and inflation mapping.Another dynamic system question which is related with lifting is call- ed mapping degree.The predecessor has also obstained many results using the above tw- o nature.Although the research of topological dynamic system may be started in the early 20th century,it has been received the widespread attention and presented bigger vigor in recent 30 years.After analyzing many references,the scientists discovered that the cover- ing space and inflation mapping were useful tool when they research the low dimension dynamic system.We obtained significant achievements by using these tools and it has attracted the general scientists'universal attention,however,the special references about the lifting and mapping degree of the two-dimensional dynamic system are not enough many mathematical models which are referred in some subjects are also two dimension- al torus self-mapping.So it is necessary to study dynamic systems of the two dimension- al torus self-mapping.The author analyses a lot of references to study the inflation mapping and mapping degree of circumference in dynamical system.This paper which is used to the covering space and lifting describes some important property of the self-mapping of torus in dynamic system(for example:lifting,mapping degree,rotation number of torus homeom- orphism,expanding mapping and so on).In this paper, we will research two questions in topological dynamics.First, Zhang Jing-zhong and Xiong Jin-cheng determined upgrade and mapping degree of continuous self-mapping f on cirle continuous S 1 in Iterative Function and Dynamical Systems of One-dimension in 1992. Considering f :T 2→T2 is a contionu- ous self-mapping of tours , using character of covering spaces and upgrade mapping,one dimensional continuous self-mapping of circle is extended to two dimensional contionu- ous self-mapping of tours.At the same time,Zhang Jing-zhong studied rotation number and expanding map of homeomorphism on cirle continuous S 1.In this paper , we debate about rotation numb- er and expanding map of homeomorphism onto itself for torus in the topological space.