The talk will discuss the dynamics of circle maps (orientation-preserving circle homeomorphisms (o.p.c.h)).
We will review the topological classification of these circle maps, namely the Poincare Classification and the Denjoy Theory. We will introduce the beautiful construction of the dynamical partition of the circle. We will review the connection between the concept of the Diophantine properties of the irrational rotation number of our map  and the smooth classification of (sufficiently regular) circle maps (the so called "rigidity" theory), in the context of three important classes: diffeomorphisms, critical maps, and maps with a break point.
For circle maps with a break point, we will discuss two recent results that were obtained in the rigidity theory of circle maps with a break point.
The first main result is a proof that C^1 rigidity holds for circle maps with a break point for almost all irrational rotation numbers. This result is joint work with Kostya Khanin and Sasa Kocic.
The second main result has to do with the family of fractional linear transformation (FLT) pairs. An FLT-pair T is a circle homeomorphism that consists of two branches each of which is an FLT. Such a map can be viewed as a circle map with two neighbouring break points lying on the same orbit. For this family, C^1 rigidity holds for all irrational rotation numbers without any restriction (the so-called "robust rigidity").