Decay of Correlations for the Rauzy-Veech-Zorich Induction Map on the Space of Interval Exchange Transformations and the Central Limit Theorem for the Teichmueller Flow on the Moduli Space of Abelian Differentials

Self-Inverses, Lagrangian Permutations and Minimal Interval Exchange Transformations with Many Ergodic Measures

Thanks to works by M. Kontsevich and A. Zorich followed by C. Boissy, we have a classification of all Rauzy Classes of any given genus. It follows from these works that Rauzy Classes are closed under the operation of inverting the permutation. In this paper, we shall prove the existence of self-inverse permutations in every Rauzy Class by giving an explicit construction of such an element satisfying the sufficient conditions. We will also show that self-inverse permutations are Lagrangian, meaning any suspension has its vertical cycles span a Lagrangian subspace in homology. This will simplify…

Dynamics and geometry of the Rauzy-Veech induction for quadratic differentials

Interval exchange maps are related to geodesic flows on translation surfaces; they correspond to the first return maps of the vertical flow on a transverse segment. The Rauzy-Veech induction on the space of interval exchange maps provides a powerful tool to analyze the Teichmueller geodesic flow on the moduli space of Abelian differentials. Several major results have been proved using this renormalization. Danthony and Nogueira introduced in 1988 a natural generalization of interval exchange transformations, namely the linear involutions. These maps are related to general measured foliations o…

对角流检测层的拓扑

Diagonal flow detects topology of strata

本文研究有根二次微分空间层分支上对角流与拓扑的关系，证明流群等于基本群及Rauzy-Veech群在环境辛群中的Zariski稠密性，从而确立了Kontsevich-Zorich猜想。

Recurrence statistics for the space of Interval Exchange maps and the Teichmüller flow on the space of translation surfaces

In this note we show that the transfer operator of a Rauzy-Veech-Zorich renormalization map acting on a space of quasi-Hölder functions is quasicompact and derive certain statistical recurrence properties for this map and its associated Teichmüller flow. We establish Borel-Cantelli lemmas, Extreme Value statistics and return time statistics for the map and flow. Previous results have established quasicompactness in Hölder or analytic function spaces, for example the work of M. Pollicott and T. Morita. The quasi-Hölder function space is particularly useful for investigating return time statisti…

区间交换变换的扭曲上循环：不变结构与 Lyapunov 谱

Twisted cocycle for interval exchange transformations: Invariant structures and Lyapunov spectrum

本文研究了区间交换变换的扭曲上循环，通过不变子丛分解和不变辛形式证明了零 Lyapunov 指数的存在性与谱对称性，并揭示了旋转型置换的谱退化现象。

Connected components of the strata of the moduli spaces of quadratic differentials

In two fundamental classical papers, Masur and Veech have independently proved that the Teichmueller geodesic flow acts ergodically on each connected component of each stratum of the moduli space of quadratic differentials. It is therefore interesting to have a classification of the ergodic components. Veech has proved that these strata are not necessarily connected. In a recent work, Kontsevich and Zorich have completely classified the components in the particular case where the quadratic differentials are given by the global square of Abelian differentials. Here we are interested in the comp…

The cohomological equation for Roth type interval exchange maps

We exhibit an explicit full measure class of minimal interval exchange maps T for which the cohomological equation $Ψ-Ψ\circ T=Φ$ has a bounded solution $Ψ$ provided that the datum $Φ$ belongs to a finite codimension subspace of the space of functions having on each interval a derivative of bounded variation. The class of interval exchange maps is characterized in terms of a diophantine condition of ``Roth type'' imposed to an acceleration of the Rauzy--Veech--Zorich continued fraction expansion associated to T. CONTENTS 0. Introduction 1. The continued fraction algorithm for interval exchange…