**Title**: Cr Closing lemma for partially hyperbolic diffeomorphisms on 3-manifolds

**Speaker**: Yi Shi (Peking University)

**Abstract**: The Cr-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 Cr-close system that has a periodic orbit passing through a given point. And this point is called Cr-closable. Steve Smale listed the Cr-closing lemma as one of mathematical problems for this century.

In this talk, we prove the Cr(r=2,3,⋯,∞) closing lemma for partially hyperbolic diffeomorphisms on 3-manifolds: every non-wandering point of these diffeomorphisms is Cr-closable. Moreover, we will show that Cr-generic conservative partially hyperbolic diffeomorphisms on 3-manifolds have dense periodic points.