Parameterized KAM Theorem Of Symplectic Mappings

This thesis discusses the existence of invariant tori in integrable exact symplectic mappings with multiple degrees of freedom under small perturba-tions. For integrable exact symplectic mapping under small perturbations, the frequencies as parameters are introduced to the generating function of the symplectic mapping. We prove that if the integrable exact symplectic mapping under small perturbations parameterized meets that the generating function is real analytic, the frequency mapping is nondegenerate and the perturbation is sufficiently small in a functional space, then the integrable exact symplectic mapping under small perturbations will be integral states in the phase space. In other words, for majority frequency parameters, the integrable exact sym-plectic mapping under small perturbations still exists invariant torus.In Chapter1, we introduce the history and research status of the KAM theory and several classical non-degenerate conditions in KAM theory(Kolmogorov-nondegenerate、IsoeiJergetic-nondegenerate、Russnmn-nondegenerate and the relationships between them). Then we introduce the research background, main content and the significance of this thesis;In Chapter2, we introduce the preliminary knowledge to analysis the existence of invariant tori of the integrable exact symplectic mapping under small perturbations, including symplectic mapping, nondegenerate condition, Diphantine condition, the Cauchy estimate, the mean value theorem and some lemmas used in the proof of our result;In Chapter3, for the integrable exact symplectic mapping under small perturbations, we introduce the method of the frequencies as parameters sep-arated from actions. Then we give the normal form of the integrable exact symplectic mapping parameterized under small perturbations and the main theorem of the thesis;In Chapter4, we give the proof of the main theorem.