I will describe recent advances in the Pugh-Shub stable ergodicity theory for partially hyperbolic diffeomorphisms. In particular, I consider two "competing" methods to show that a given partially hyperbolic diffeomorphism is stably ergodic (i.e., it is ergodic along with any of its sufficiently small perturbations). One of them relates the problem to to the global estimates of the action of the system along its central direction while another one deals with a more delicate estimates using Lyapunov exponents in the central direction.

In this talk, we will begin with simple questions of the Diophantine Approximation Theory, for instance, how closely can a given irrational number x be approximated by a rational number r with denominator no larger than a fixed number? This will lead us to talk about the set known as the Lagrange Spectrum whose structure closely resembles the structure of the sum of dynamically defined Cantor sets, which are defined by an iterate system of expanding differentiable functions on intervals. We will construct two Cantor sets whose arithmetic sum is a uniformly contracting self-similar set. A local result on a sufficient condition for a uniformly contracting self-similar set to be of Lebesgue measure zero will be proven.

We will review the recent (and not so recent) results on dynamics of piecewise isometries (especially piecewise translations), both in one and in higher dimensional case. Some interesting results (by Suzuki, Goetz, Zhuravlev, Boshernitzan, Bruin, Troubetzkoy, Buzzi) are known, but most of natural questions are still open. The main goal of the talk is to expose these open questions to potential researchers.