School and Conference in Dynamical Systems 2012 (ICTP, Trieste, Italy)

It started this week the 2012 edition of the (by now traditional) School and Conference in Dynamical Systems at ICTP (Trieste, Italy). This year, it is organized by Stefano Luzzatto, Marcelo Viana and Jean-Christophe Yoccoz and its format is as follows.

The first two weeks (i.e., from May 21 to June 1) are the “school part”. In particular, there are 3 minicourses (each of 45 min. per day over 10 days):

1. “Markov partitions for surface diffeomorphisms” by Krerley Oliveira and Omri Sarig,
2. “Dynamics of partially hyperbolic systems and rigidity” by Pablo Carrasco and Federico Rodriguez-Hertz, and
3. “Birkhoff sums for interval exchange maps: the Kontsevich-Zorich cocycle” by Jean-Christophe Yoccoz and myself.

The last week (i.e., from June 4 to June 8) is the “conference part” with research talks by dynamicists from several countries.

One can find at the bottom part of this link here a dynamical schedule of the event (who is kept as updated as possible by Madame Koutou Mabilo), and lecture notes/slides of the 3 minicourses (and some research talks occurring in the intervals between minicourses).

The material covered in minicourse #3 (with links for the available material) was:

1. May 21: Rotations on the circle : rational vs. irrational dichotomy and unique ergodicity (after a quick review of invariant measures, Birkhoff sums and Birkhoff’s theorem); Linear flows on the 2-torus : rotations of as return maps of linear flows, rational vs. irrational dichotomy for linear flows. (See J.-C. Yoccoz’s slides here).
2. May 22: Rotations as interval exchange maps; Renormalization algorithm: slow and fast versions; Continued fraction algorithm and Gauss measure; Birkhoff sums for Diophantine rotations; Suspension of rotations and linear flows; Geodesic flow on the modular surface and its relation to continued fraction algorithm. (See J.-C. Yoccoz’s slides here).
3. May 23: Definition of interval exchange transformations (i.e.t.’s); Combinatorial data, irreducibility, and length data; Matrix ; Suspensions of i.e.t.’s; Translation surfaces; Genus, number of marked points, homology/cohomology (and matrix ) of suspensions of i.e.t.’s. (See my slides here).
4. May 24: General comments on renormalization dynamics; Connections for i.e.t.’s and translation flows; Minimality and Keane’s theorem; Basic step of Rauzy-Veech algorithm for i.e.t.’s; Rauzy diagrams; Complete paths in Rauzy diagrams; Basic step of Rauzy-Veech algorithm for suspensions of i.e.t.’s. (See my slides here).
5. May 25: Quick review of previous lecture; Iteration of RV algorithm for i.e.t.’s; Evolution of length and translation data under RV algorithm and matrix ; Matrix , discrete version of KZ cocycle and its relation with special Birkhoff sums; Symplecticity of (restricted version of) ; Sketch of application of Keane’s conjecture on unique ergodicity of i.e.t.’s; Definition of Teichmuller spaces of translation surfaces; Examples of translation surfaces of genus 1 defining the same point in Teich. space. (See my slides here).
6. May 28: Teichmuller and moduli spaces of translation surfaces; period coordinates; Masur-Veech measures; -action on the Teichmuller and moduli spaces of unit area translation surfaces; Teichmuller flow; Statement of Masur-Veech’s theorem on the finiteness of Masur-Veech measure and ergodicity (and mixing) of Teichmuller flow on connected components of strata of moduli spaces of unit area translation surfaces. (See J.-C. Yoccoz’s slides here)
7. May 29: Quick review of previous lecture; Comparison between Teichmuller flow and Rauzy-Veech algorithm via Veech boxes on moduli spaces; Definition of the continuous time version of Kontsevich-Zorich (KZ) cocycle; Comparison between the continuous time version of KZ cocycle and discrete time versions (matrix over Rauzy-Veech algorithm). (See J.-C. Yoccoz’s slides here)
8. May 30: Zorich’s phenomenon on deviations of ergodic averages for i.e.t.’s and its explanation via Lyapunov exponents of KZ cocycle; Kontsevich-Zorich conjecture on the simplicity of Lyapunov spectra of Masur-Veech measures, Forni’s theorem and Avila-Viana theorem; General comments on the proof of Avila-Viana theorem and statement of Avila-Viana simplicity criterium. (See J.-C. Yoccoz’s slides here)
9. May 31: General review showing the ”big picture” of the course (so far), and review of the statement of Avila-Viana theorem; Reduction of Avila-Viana simplicity criterium to the detection of adequate cocycle-invariant sections on  Grassmannians (Theorem 1); Reduction of Theorem 1 to show the convergence (under the cocycle) of prob. on Grassmannians towards Dirac masses (Prop. 3) via the notion of u-states and the martingale convergence theorem. (See my slides here)
10. June 1: End of proof of Prop. 3 (convergence under the cocycle to Dirac masses on Grass.) using pinching and twisting properties; Statement of Eskin-Kontsevich-Zorich formula for sums of Lyap. exp. of KZ cocycle wrt general -inv. prob. modulo Siegel-Veech constants; Discussion on Siegel-Veech constants in two settings: genus 2 transl. surf. and square-tiled surfaces; Computation with EKZ formulas in some concrete examples; Delecroix-Hubert-Lelievre’s work on Ehrenfest wind-tree model as a ”physical” application of knowledge of explicit values of Lyap. exp. of KZ cocycle. (See my slides here)

Finally, let me give a list of useful references containing complete proofs of facts skipped during the lectures and/or discussions of topics omitted from the minicourse due to time limitations:

• Continued fraction algorithms for interval exchange maps: an introduction (by J.-C. Yoccoz)
• Interval exchange maps and translation surfaces (by J.-C. Yoccoz)
• Dynamics of interval exchange maps and Teichmüller flows (by M. Viana)
• Flat surfaces (by A. Zorich)
• Rational billiards and flat structures (by H. Masur and S. Tabachnikov)
• SPCS 6 (for more discussion on Avila-Viana simplicity criterium)