In 2007, the Clay Math. Institute sponsored a Summer School on Homogenous Flows, Moduli Spaces and Arithmetic. A nice feature of this summer school was the fact that the speakers were invited to provide lecture notes of their mini-courses. In particular, a few days ago, J. C. Yoccoz made his lectures notes publicly available at his website (see here) because he also plans to use them for his forthcoming 2009-2010 course at College de France (see Yoccoz homepage or this previous post).

Durante as quartas-feiras do periodo de 18 de Novembro de 2009 ate 3 de Fevereiro de 2010, o professor Jean-Christophe Yoccoz do College de France (CDF) ira ministrar das 9 as 11 da manha (na filial do CDF localizada no numero 3 da Rue d’Ulm) o curso intitulado:

Renormalisation des échanges d’intervalles généralisés

Segundo as informacoes que o Jean-Christophe Yoccoz me passou, a ideia sera falar de um trabalho ainda em andamento com Stefano Marmi sobre as chamadas transformacoes de intercambios de intervalos generalizadas. Mais precisamente, o objetivo deles e entender uma certa dinamica de renormalizacao no espaco de tais transformacoes no sentido de dar continuidade a um trabalho anterior de Stefano Marmi, Pierre Moussa e o proprio Jean-Christophe sobre as transformacoes de intercambios de intervalos afins.

Alem disso, a Aline pretende tomar notas deste curso, de modo que, na medida em que o tempo permitir, provavelmente iremos escanear e postar aos poucos estas notas.

Hi! Firstly, let me apologize for the 1 month and half hiatus: while I plan to close up my series on the talks and courses of the Dynamical Systems conference (held at ICTP, Italy), I wasn’t able to do so during April because I was learning some interesting facts about square-tiled surfaces and Teichmuller dynamics from my new friends Pascal Hubert and Anton Zorich. Of course, I plan to share these facts in some future posts, although I can’t promise to do so in the next few months. On the other hand, I can give a clue to the curious reader by providing a copy of the slides of my recent talk at Orsay entitled: Sur le cocycle de Kontsevich-Zorich au-dessus de deux origamis spéciaux.

Warning. While the title is written in French (and the actual talk was my third presentation speaking in French), the slides are written in English (because I don’t feel comfortable to write in French yet).

Before giving the link to the slides, let me make a brief description of the talk.

The basic objective was the discussion of the Teichmuller geodesic flow and the Kontsevich-Zorich cocycle. Nowadays, after the works of Forni (2002), Avila and Viana (2006), we know that the Kontsevich-Zorich conjecture is true: the Lyapunov spectrum of the Kontsevich-Zorich (KZ) cocycle with respect to the absolutely continuous -invariant ergodic probability is non-uniformly hyperbolic (i.e., 0 doesn’t belong to the spectrum) and simple (i.e., all exponents have multiplicity one). In other words, the ergodic behavior of generic points with respect to the Lebesgue measure is more or less well-understood.

However, one can follow Veech and pose the question: what about the ergodic behavior of non-generic points? In this direction, Forni (2005) and Forni, Matheus (2008) showed that there are examples of orbits with totally degenerate Kontsevich-Zorich  spectrum (see this previous post). An interesting feature of the KZ cocycle over totally degenerate orbits is the fact that it is isometric (this follows from Forni’s 2002 work). In the talk, I discussed an improvement of this fact (obtained jointly with Jean-Christophe Yoccoz): by computing the action of the group of affine diffeomorphisms on the relative homology of these examples, we are able to show that the Kontsevich-Zorich cocycle is represented by a subgroup of the Weyl group of the root system . A nice consequence is the fact that the KZ cocycle acts by a finite group of rigid matrices, so that this gives a new proof of Forni and Matheus results by more explicit methods.

Finally, the slides can be found here.

Escola de Altos Estudos

Tópicos recentes em Sistemas Dinâmicos

IMPA, Rio de Janeiro, de 05 a 30 de janeiro de 2009.

Este curso foi ministrado pelo Professor Jean-Christophe Yoccoz , Collège de France, no âmbito da Escola de Altos Estudos da Capes.

As aulas aconteceram no IMPA e foram transmitidas pela internet para todo o Brasil.

♦ As notas e os artigos relacionados com a primeira semana do curso são:

• Echanges d’intervalles (notas em francês da teoria de Intercambio de Intervalos)
• Continued Fraction Algorithms for Interval Exchange Maps : an Introduction (notas em inglês da teoria de Intercambio de Intervalos)
• Affine interval exchange maps with a wandering interval (artigo publicado junto com S. Marmi e P. Moussa)
  
• Echanges d`intervalles et surfaces de translation (notas do seminário Bourbaki)

Vídeos (este material pode ser copiado desde que seja para fins educativos. Fica proibida a venda) :

• 05-01-2009 – Segunda Feira

• 07-01-2009 – Quarta Feira

• 09-01-2009 – Sexta Feira

Minhas anotações feitas na primeira parte do curso clique aqui !

♦♦ As notas e os artigos relacionados com a segunda semana do curso são:

• Some questions and remarks about SL(2, R) cocycles (artigo elaborado por J-C Yoccoz em homenagem a Anatole Katok)
• Uniformly Hyperbolic Finite-Valued SL(2,R)-Cocycles (artigo de A. Avila, J. Bochi e J-C Yoccoz)

Vídeos (este material pode ser copiado desde que seja para fins educativos. Fica proibida a venda) :

• 12-01-2009 – Segunda Feira

• 14-01-2009 – Quarta Feira

• 16-01-2009 – Sexta Feira

• 19-01-2009 – Segunda Feira

Minhas anotações feitas na segunda parte do curso clique aqui !

♦♦♦ As notas e os artigos relacionados com a terceira e quarta semana do curso são:

• Non-Uniformly Hyperbolic Horseshoes Arising from Bifurcations of Poincaré Heteroclinic Cycles. (artigo publicado junto com J. Palis)
• Fers à cheval non uniformément hyperboliques engendrés par une bifurcation homocline et densité nulle des attracteurs (artigo elaborado junt com J. Palis)

Vídeos (este material pode ser copiado desde que seja para fins educativos. Fica proibida a venda) :

• 21-01-2009 – Quarta Feira

• 23-01-2009 – Sexta Feira

• 26-01-2009 – Segunda Feira

• 28-01-2009 – Sexta Feira

• 30-01-2009 – Segunda Feira

Minhas anotações feitas na terceira parte do curso clique aqui !

No Summer School 2007 in Pisa foram ministrados excelentes cursos. Aos poucos, as notas desses cursos estão sendo colocadas  para acesso de todos. Algumas delas ja estao disponiveis:

• Equidistribution and L-Functions

Gergely Harcos

• Reveiw of Vatsal’s work on equidistribution and non-vanishing L-functions

Nicolas Templier

• Fuchsian groups, geodesic flows on surfaces of constant negative curvature and symbolic coding of geodesics

Svetlana Katok

• Eigenfunctions of the laplacian : a semiclassical study

Nalini Anantharaman

Os textos abaixo se encontram na Homepage de Jean-Christophe Yoccoz.

1. Examples of compact hypersurfaces
2. Sur une question de Jacob Palis
3. Sur une proposition d’Oxtoby
4. Non existence of Lagrangian graphs
5. Conjugaison quasi symetrique des homéomorphismes analytiques du cercle à des rotations
6. Uniformité de la distorsion de swiatek pour les familles compactes de produits de blaschke
   

E no site se encontram também material dos cursos por ele dados:

1. Cocycles uniformément hyperboliques
2. Echanges d’intervalles
3. Continued Fraction Algorithms for Interval Exchange Maps : an Introduction