Exponential decay of correlations for the Rauzy-Veech-Zorich induction map

无限型IET的Rauzy-Veech归纳

Rauzy-Veech Induction for Infinite-Type IETs

针对来自无限亏格有限面积平移曲面的无限型区间交换变换，推广Rauzy-Veech归纳并证明在ℓ¹拓扑下唯一遍历性在参数集中稠密。

Deducing Three Gap Theorem From Rauzy-Veech Induction

The Three Gap Theorem states that there are at most three distinct lengths of gaps if one places $n$ points on a circle, at angles of $z, 2z, 3z, \ldots nz$ from the starting point. The theorem was first proven in 1958 by Sós and many proofs have been found since then. In this note we show how the Three Gap Theorem can easily be deduced by using Rauzy-Veech induction.

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

A stretched-exponential bound is obtained for the decay of correlations of the Rauzy-Veech-Zorich induction map on the space of interval exchange transformations. A Corollary is the Central Limit Theorem for the Teichmueller flow on the moduli space of abelian differentials. The proof involves an approximation of the induction map by a Markov chain satisfying the Doeblin condition, the method of Sinai and Bunimovich--Sinai. The main estimate is Lemma 4. As a corollary of such "loss of memory" estimates, integrability of a small power of the Kontsevich-Zorich cocycle is obtained, thus yielding …

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…

亏格4中一个对称折痕的算术Kontsevich--Zorich单值群

An arithmetic Kontsevich--Zorich monodromy of a symmetric origami in genus 4

本文证明了亏格4中存在一个对称折痕，其Kontsevich--Zorich单值群在Sarnak意义下是算术的，揭示了折痕的Veech群与单值群之间缺乏关联。

Decoding Rauzy Induction: An Effective Answer to Bufetov's Question

A typical interval exchange transformation has an infinite sequence of matrices associated to it by successive iterations of Rauzy induction. In 2010, W. A. Veech answered a question of A. Bufetov by showing that the interval exchange itself may be recovered from these matrices and must be unique up to topological conjugation. In this work, we will improve upon these results by providing an algorithm to determine the initial transformation from a sufficiently long finite subsequence of these matrices. We also show the defined length to be necessary by constructing finite sequences of Rauzy ind…

Large Genus Asymptotics for Siegel-Veech Constants

In this paper we consider the large genus asymptotics for two classes of Siegel-Veech constants associated with an arbitrary connected stratum $\mathcal{H} (α)$ of Abelian differentials. The first is the saddle connection Siegel-Veech constant $c_{\text{sc}}^{m_i, m_j} \big( \mathcal{H} (α) \big)$ counting saddle connections between two distinct, fixed zeros of prescribed orders $m_i$ and $m_j$, and the second is the area Siegel-Veech constant $c_{\text{area}} \big( \mathcal{H}(α) \big)$ counting maximal cylinders weighted by area. By combining a combinatorial analysis of explicit formulas of …

Decoding Rauzy Induction: An Answer to Bufetov's General Question

Given a typical interval exchange transformation, we may naturally associate to it an infinite sequence of matrices through Rauzy induction. These matrices encode visitations of the induced interval exchange transformations within the original. In 2010, W. A. Veech showed that these matrices suffice to recover the original interval exchange transformation, unique up to topological conjugacy, answering a question of A. Bufetov. In this work, we show that interval exchange transformation may be recovered and is unique modulo conjugacy when we instead only know consecutive products of these matri…

Simplicity of the Lyapunov spectra of certain quadratic differentials

We prove that the "plus" Rauzy--Veech groups of the connected components of all strata of meromorphic quadratic differentials defined on Riemann surfaces of genus at least one having at most simple poles and at least three singularities (zeros or poles), not all of even order, are finite-index subgroups of their ambient symplectic groups. This shows that the "plus" Lyapunov spectrum of such strata is simple. Moreover, we show that the index of the "minus" Rauzy--Veech group is also finite for connected components of strata satisfying the same conditions and having exactly two singularities of …

$\mathcal{H}(2)$ 中折纸的 Kontsevich-Zorich 单值群的指标

Index of the Kontsevich-Zorich monodromy of origamis in $\mathcal{H}(2)$

本文研究了 $\mathcal{H}(2)$ 层中 origamis 的 Kontsevich-Zorich 单值群，证明了其在 $SL_2(\mathbb{Z})$ 中的指标仅为 1 或 3。

Large Genus Asymptotics for Volumes of Strata of Abelian Differentials

In this paper we consider the large genus asymptotics for Masur-Veech volumes of arbitrary strata of Abelian differentials. Through a combinatorial analysis of an algorithm proposed in 2002 by Eskin-Okounkov to exactly evaluate these quantities, we show that the volume $ν_1 \big( \mathcal{H}_1 (m) \big)$ of a stratum indexed by a partition $m = (m_1, m_2, \ldots , m_n)$ is $\big( 4 + o(1) \big) \prod_{i = 1}^n (m_i + 1)^{-1}$ as $2g - 2 = \sum_{i = 1}^n m_i$ tends to $\infty$. This confirms a prediction of Eskin-Zorich and generalizes some of the recent results of Chen-Moeller-Zagier and Sauva…