Title: $C^r$ Closing lemma for partially hyperbolic diffeomorphisms on 3-manifolds
Speaker: Yi Shi (Peking University)
Abstract: 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. And this point is called $C^r$-closable. Steve Smale listed the $C^r$-closing lemma as one of mathematical problems for this century.
In this talk, we prove the $C^r (r=2,3,\cdots, \infty)$ closing lemma for partially hyperbolic diffeomorphisms on 3-manifolds: every non-wandering point of these diffeomorphisms is $C^r$-closable. Moreover, we will show that $C^r$-generic conservative partially hyperbolic diffeomorphisms on 3-manifolds have dense periodic points.
This is a joint work with Shaobo Gan.