Convergence Of Normalization Transformations For Symplectic Mappings Under The Brjuno Condition

In order to study the behaviour of the given symplectic mapping, one usually reduce the original system to the simplest normal form by coordinate transformations. However, the normalization transformations and the normal form are all divergent generally. Only when the normal form is a special one and the eigenvalues satisfy the Diophantine or Brjuno condition, are the normalization transformations convergent.The thesis mainly studies the convergence of normalization transforma-tions for symplectic mappings under the Brjuno condition. It is divided into four chapters.In Chapter1, We introduce the development history of the normal form theory, symplectic mappings and main content of the thesis.In Chapter2, We introduce the preliminaries which are used to prove the main result in the thesis.In Chapter3, the main result is stated. We normalize symplectic map-pings by the1-time mapping of Hamiltonian systems, and obtain that the composition of normalization transformations is convergent if the normal form is linear and eigenvaues of the linear system satisfy the Brjuno condition.In Chapter4, the proof of our main result is given. This chapter is divided into three parts. The first part is to construct the iteration transformation. In the second part, we estimate the iteration transformation and the new re-mainder. The third part is to prove the convergence of transformation under the Brjuno condition.