报告人 史逸（研究员）（北京大学）


报告时间 2018年3月9日09:30-11:30

报告地点  精正楼二楼学术报告厅

报告摘要  The $C^r$ closing lemma is one well-known problem in the theory of dynamical systems. The problem is to perturb the original dynamical system so as to obtain a $C^r$-close system that has a periodic orbit passing through a given point. 

This problem arose from Poincar/'e's famous paper "New Methods in Celestial Mechanics", which studies the existence of periodic solutions of Three-Body Problem. After more than fifty years, Thom asked the problem for general diffeomorphisms and vector fields. In 1967, Pugh solved this problem for $r=1$. However, it turns out that the "local perturbation" techniques for solving this problem in $C^1$-topology will not work for $r/geq2$, and the perturbation must be global. Smale list this problem for $r/geq2$ as one of the mathematical problems in this century.

In this talk, we will prove the $C^r$ $(r=2,3,/cdots,/infty)$ closing lemma for conservative partially hyperbolic diffeomorphisms on 3-manifolds. This shows that for $C^r$-generic conservative partially hyperbolic diffeomorphisms on 3-manifolds have dense periodic points. If time permits, we will also discuss this problem for general partially hyperbolic diffeomorphisms with one-dimensional center.

This is a joint work with Shaobo Gan.