**Date & Time:** Thursday, February 20, 2014, 14:30-15:30.

**Venue:** Ramanujan Hall

**Speaker:** Konstantin Khanin, University of Toronto

**Title:** Renormalization and Rigidity in Circle Dynamics

**Abstract:** Linearization of circle diffeomorphisms with
irrational rotation numbers is a classical problem
which goes back to the works of V. Arnold and
M. Herman. It can be viewed as the one-dimensional
KAM (Kolmogorov-Arnold-Moser) theory.

It turns out that a parallel theory can be constructed for circle maps with singularities. In this case smooth linearization is impossible. However in many cases one can prove rigidity results. Namely, two maps with the same type of singularity and the same irrational rotation number are smoothly conjugate to each other, provided that certain regularity conditions are met. The main tool in establishing rigidity results is a convergence of renormalizations. In this talk we shall present results on rigidity and renormalization in three cases: smooth diffeomorphisms, critical circle maps, and circle maps with breaks.