Metric entropy, KAM theory and dual lens maps

In 1950s Kolmogorov asked the following question, which is closely related to the celebrated KAM theory: Can a non-degenerate nearly integrable Hamiltonian system have a positive Kolmogorov-Sinai entropy (a.k.a. metric entropy)? We give a positive answer to this question.
In fact, examples with positive entropy can be constructed by perturbing the standard geodesic flow on the 3-dimensional torus. And in higher dimensions there are examples with more interesting properties. The constructions are based on dual lens maps, which determine the relation between incoming and outgoing directions of geodesics passing through a given region. In the talk I will give an introduction to the background of the problem as well as of the theory of dual lens maps, and will explain some of the ideas behind the proof. It is a joint work with D. Burago and S. Ivanov.