Seminar 1.17.19 Shi | Ergodic Theory at Ohio State

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 $C^{r} (r = 2, 3, \dots, \infty)$ 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.