The Teichmüller space of an Anosov diffeomorphism of $T^2$

Gromov双曲空间与Teichmüller空间上随机等距变换平移长度的线性增长

Linear growth of translation lengths of random isometries on Gromov hyperbolic spaces and Teichmüller spaces

该研究针对Gromov双曲空间等距群上的随机游走，证明无矩条件下非初等随机游走平移长度至少线性增长，有限一阶矩下增长率等于漂移，还将技术推广到Teichmüller空间，在有限一阶矩下证明谱定理，拓展了已有结果。

Lagrange Spectra in Teichmüller Dynamics via renormalization

We introduce Lagrange Spectra of closed-invariant loci for the action of SL(2,R) on the moduli space of translation surfaces, generalizing the classical Lagrange Spectrum, and we analyze them with renormalization techniques. A formula for the values in such spectra is established in terms of the Rauzy-Veech induction and it is used to show that any invariant locus has closed Lagrange spectrum and values corresponding to pseudo-Anosov elements are dense. Moreover we show that Lagrange spectra of arithmetic Teichmüller discs contain an Hall's ray, giving an explicit bound for it via a second for…

子流形映射类群的长度谱非算术性

Non-arithmeticity of length spectra of subgroups of mapping class groups

本文证明了非初等子群的Teichmüller长度谱非算术，通过引入MF和PMF上的交叉比概念，研究其几何和动力学性质，揭示了Teichmüller空间和PMF上的非负曲率特征缺失下的结果。

Meromorphic $L^2$ functions on flat surfaces

We prove a quantitative version of the non-uniform hyperbolicity of the Teichmüller geodesic flow. Namely, at each point of any Teichmüller flow line, we bound the infinitesimal spectral gap for variations of the Hodge norm along the flow line in terms of an easily estimated geometric quantity on the flat surface, which is greater than or equal to the flat systole. As applications, we strengthen results of Treviño and Smith regarding unique ergodicity of measured foliations, and give an estimate for the spectral gaps of pseudo-Anosov homeomorphisms based on the location of their axes in the mo…

The Teichmüller space of the standard action of $SL(2,Z)$ on $T^2$ is trivial

The group $SL(n,{\bf Z})$ acts linearly on $\R^n$, preserving the integer lattice $\Z^{n} \subset \R^{n}$. The induced (left) action on the n-torus $\T^{n} = \R^{n}/\Z^{n}$ will be referred to as the ``standard action''. It has recently been shown that the standard action of $SL(n,\Z)$ on $\T^n$, for $n \geq 3$, is both topologically and smoothly rigid. That is, nearby actions in the space of representations of $SL(n,\Z)$ into ${\rm Diff}^{+}(\T^{n})$ are smoothly conjugate to the standard action. In fact, this rigidity persists for the standard action of a subgroup of finite index. On the oth…

Geometry of the Complex of Curves I: Hyperbolicity

The Complex of Curves on a Surface is a simplicial complex whose vertices are homotopy classes of simple closed curves, and whose simplices are sets of homotopy classes which can be realized disjointly. It is not hard to see that the complex is finite-dimensional, but locally infinite. It was introduced by Harvey as an analogy, in the context of Teichmuller space, for Tits buildings for symmetric spaces, and has been studied by Harer and Ivanov as a tool for understanding mapping class groups of surfaces. In this paper we prove that, endowed with a natural metric, the complex is hyperbolic in …

测地线拉伸、压力度量与标记长度谱刚性

Geodesic stretch, pressure metric and marked length spectrum rigidity

本文通过测地线拉伸和方差一致估计，为标记长度谱刚性提供了新证明，并引入与Weil-Peterson度量相关的压力度量。

曲线复形上的极小渐近平移长度与映射环面的同调

Minimal asymptotic translation lengths on curve complexes and homology of mapping tori

本文研究了曲线复形上极小渐近平移长度的上下界，并构造了非Torelli的pseudo-Anosov映射类以探讨其在亏格趋于无穷时的平移长度衰减限制。