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(Universite de Cergy-Pontoise)


Title: Reducibility and almost reducibility: in quasi-periodic dynamics: a renormalization approach.


 I will discuss different notions of reducibility (reducibility, almost reducibility, accumulation by reducible systems)  for various quasi-periodic dynamical systems such as diffeomorphisms of the circle, quasi-periodic cocycles and pseudo-rotations of the disk. After a short introduction to KAM theory, that applies in  local (perturbative) situations, I will describe the renormalization method  that allows to tackle global (non perturbative) problems. 


Course 1:  9月1日14:00-15:30 蒙民伟楼1105

Title:  Introduction to reducibility and related concepts for various quasi-periodic systems. 

Abstract: A short presentation of perturbative techniques (KAM)  and of the idea of  renormalization for  diffeomorphisms of the circle, quasi-periodic cocycles and pseudo-rotations of the disk


Course 2: 9月1日15:30-17:00 蒙民伟楼1105

Title: Reducibility of smooth diffeomorphisms of the circle.

Anstract: I will introduce: Rotation number, Denjoy Theorem, Arnold's theorem, Herman-Yoccoz theorem on linearization, Yoccoz's accumulation result and the notion of almost-reducibility.


Course 3: 9月1日18:00-19:30 蒙民伟楼1105

Title: A proof of Herman-Yoccoz theorem via renormalization.

Abstract: I'll present a proof using renormalization of the famous result of Herman and of Yoccoz on linearization of smooth circle diffeomorphisms.


Course 4: 9月1日19:30-21:00 蒙民伟楼1105

Title: Almost reducibility of circle diffeomorphisms.

Abstract: I will present a proof of almost-reducibility of circle diffeomorphisms (joint with A. Avila): any smooth  circle diffeomorphism with an irrational rotation number can be conjugated to a smooth diffeomorphism arbitrarily close to a rotation. 


Course 5: 9月2日9:00-10:30 蒙民伟楼1105

Title: Renormalization of quasi-periodic cocycles.

Abstract: I will present the theory of renormalization of quasi-periodic cocycles and discuss some application to the spectral theory of quasi-periodic Schr"odinger operators.



Course 6: 9月2日10:30-12:00 蒙民伟楼1105

Title: On the almost-reducibility of pseudo-rotations of the disk. 

Abstract: A pseudo rotation of the disk is a smooth diffeomorphism of the disk preserving (globally) the boundary of the disk, area preserving, and with only one periodic point (the origin). Besides the conjugates to rigid rotations, examples of pseudo rotations are the Anosov-Katok (weak mixing) examples. A natural question is to understand whether these two classes of examples are the only one. An important notion in this problem is that of almost-reducibility. I will present a result we proved with A. Avila in the semi-local case.