Ergodic Theory / Probability Seminar - Yi Shi

January 12, 2019
Thursday, January 17, 2019 - 3:00pm to 4:00pm
Math Tower 154
Ergodic Theory/Probability Seminar

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.

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