ICTP Mathematics Seminar - Thursday 20 June 2013, at 16:00

[Mabilo]

M A T H E M A T I C S   S E M I N A R S   2013

Thursday, 20 June, at 16.00 hrs.

Circle Maps, Renormalization, and Rigidity Theory

Elio Mazzeo
(University of Toronto, Canada)

Abstract:
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").

VENUE: Luigi Stasi Seminar Room, first floor , Leonardo Building