近可积哈密顿系统的低维作用极小测度.docVIP

近可积哈密顿系统的低维作用极小测度 王楷植, 李勇 吉林大学数学学院，长春　 130012 摘要：本文发展了一种参数化的弱 KAM 技巧, 它可以被视为弱 KAM 理论中的部分结果从有 限维到无穷维的一种推广. 在此框架下, 将上述技巧局部地应用于近可积凸哈密顿系统, 我们得 到了低维作用极小测度的存在性. 支撑作用极小测度的低维不变集是低维不变环面的推广. 进 一步, 我们尝试将主要结果推广至非凸情形. 在弱于严格凸性的条件下, 我们仍得到了近可积哈 密顿系统低维作用极小测度的存在性. 关键词：哈密顿系统, 低维作用极小测度, 近可积哈密顿系统, 弱 KAM 理论 中图分类号： O175.13 Lower dimensional action minimizing measures for nearly integrable Hamiltonian systems WANG Kai-Zhi, LI Yong College of Mathematics, Jilin University, Changchun 130012 Abstract: In this article, we develop a parametrized weak KAM technique, which can be regarded as a generalization from the ?nite dimensional case to the in?nite dimensional case of partial results in the weak KAM theory. In such a framework, applying the technique to the nearly integrable convex Hamiltonian systems locally, we obtain the existence of lower dimensional action minimizing measures. The lower dimensional invariant sets, which support the action minimizing measures, are generalizations of lower dimensional invariant tori. Furthermore, we attempt to generalize our main result to the nonconvex case. Under certain weaker conditions than strict convexity, we still provide an existence result of lower dimensional action minimizing measures for the nearly integrable Hamiltonian systems. Key words: Hamiltonian systems, lower dimensional action minimizing measures, nearly integrable Hamiltonian systems, weak KAM theory 0 Introduction The present work concerns the existence of lower dimensional action minimizing measures in nearly integrable Hamiltonian systems. More precisely, we study smooth perturbations of 基金项目： Specialized Research Fund for the Doctoral Program of Higher Education (20100061120094, 20090061110001 ). 作者简介： Wang Kaizhi (1981-), male, associate professor, major research direction：Hamiltonian systems. Li Yong (1958-) male, professor, major research direction：Hamiltonian systems. Email: kaizhiwang@jlu.edu.cn, liyong@jlu.edu.cn -1- an integrable Hamiltonian system in a given resonant