Lucien Clavier
We investigate specific examples of locally-defined real vector-fields on strata of translation surfaces. Integrating SL(2,R)-loci of Veech surfaces along these vector-fields yield interesting new examples of horocyle-invariant ergodic measures. These measures are supported on closed immersed manifolds with boundary that approach the boundary of strata in a novel way.
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On the Space of Ergodic Measures for the Horocycle Flow on Strata of Abelian Differentials
本文研究平移曲面分层$\mathcal{H}(2)$上的水平流，证明存在一列支撑在周期轨道上的遍历测度弱收敛到非遍历不变测度，并由此得出某些点的水平流轨道不向任何不变测度等分布。
Classification of horocycle orbit closures in $ \mathbb{Z} $-covers
本研究完整分类紧双曲曲面$\mathbb{Z}$-覆叠中的 horocycle 轨道闭包，通过分析覆叠中距离最小化测地射线的效率，证明非极大轨道闭包虽呈分形特征却具有整数 Hausdorff 维数。
We show that the Hausdorff dimension of any proper Teichmüller horocycle flow orbit closure on any $\mathrm{SL}{(2,\mathbf{R})}$-invariant subvariety of Abelian or quadratic differentials is bounded away from the dimension of the subvariety in terms of the polynomial mixing rate of the Teichmüller horocycle flow on the subvariety. The proof is based on abstract methods for measurable flows adapted from work of Bourgain and Katz on sparse ergodic theorems.
Tremors and horocycle dynamics on the moduli space of translation surfaces
本文在平移曲面层上引入震颤形变，并借此构造出 horocycle 流轨道的新行为，包括支撑严格包含于轨道的通用测度、非通用轨道以及 Hausdorff 维数非整数的轨道闭包。
The orbit closure of any translation surface under the horocycle flow in almost any direction equals its $SL_2(\mathbb{R})$ orbit closure. This result gives rise to new characterizations of lattice surfaces in terms of the hororcycle flow.
We study dynamics of the horocycle flow on strata of translation surfaces, introduce new invariants for ergodic measures, and analyze the interaction of the horocycle flow and real Rel surgeries. We use this analysis to complete and extend results of Calta and Wortman classifying horocycle-invariant measures in the eigenform loci. We classify the orbit-closures and prove that every orbit is equidistributed in its orbit-closure. We also prove equidistribution statements regarding limits of sequences of measures, some of which have applications to counting problems.
Minimizing laminations in regular covers, horospherical orbit closures, and circle-valued Lipschitz maps
本文揭示了紧双曲流形的$\mathbb{Z}$-覆盖中距离极小叶状结构与球面轨道闭包的联系，构造了具有指定性质的$\mathbb{Z}$-覆盖并明确描述了所有horocycle轨道闭包，同时证明双曲度量的微小扰动会导致其拓扑发生剧烈变化。
Horocycles in hyperbolic 3-manifolds with round Sierpiński limit sets
本文对具有圆形谢尔宾斯基极限集的几何有限双曲三维流形，分类了标架丛上水平圆流的轨道闭包，证明水平圆闭包为恰当浸入子流形，推广了凸余紧情形下的已知结果。
The orbit closure of the unfolding of every rational right and isosceles triangle is computed and the asymptotic number of periodic billiard trajectories in these triangles is deduced. This follows by classifying all orbit closures of rank at least two in hyperelliptic components of strata of Abelian and quadratic differentials. Additionally, given a fixed set of angles, the orbit closure of the unfolding of all unit area rational parallelograms, isosceles trapezoids, and right trapezoids outside of a discrete set is determined.
The object of this paper is to study GL(2,R) orbit closures in hyperelliptic components of strata of abelian differentials. The main result is that all higher rank affine invariant submanifolds in hyperelliptic components are branched covering constructions, i.e. every translation surface in the affine invariant submanifold covers a translation surface in a lower genus hyperelliptic component of a stratum of abelian differentials. This result implies finiteness of algebraically primitive Teichmuller curves in all hyperelliptic components for genus greater than two. A classification of all GL(2…
Topology of horocycles on geometrically finite nonpositively curved surfaces
该研究针对有限生成基本群的秩1非正曲率曲面，探究horocycle的拓扑性质，完成半horocycle分类、确定horocycle流非游荡集并刻画存在极小集的曲面，推进了相关动力系统与几何拓扑研究。
Quasidiagonals in Strata of Translation Surfaces
本文研究平移曲面分层乘积中的 GL(2,R) 轨道闭包，给出其分类，并将结果用于分析特定 Masur-Veech 测度的 joinings。
Every GL(2,R)-orbit in hyperelliptic components of strata of abelian differentials in genus greater than two is either closed, dense, or contained in a locus of branched covers.
Components of strata of k-differentials and their orbit closures
本文研究k-微分流形strata的组成部分分类，证明在高亏格情况下，除了例外情况外，非双曲分量仅有一个，且通过Arf不变量区分。同时，证明了正亏格strata的最小GL(2,R)-轨道闭包的最大性，回答了Mirzakhani-Wright的问题。
We compute a closed formula for the class of the closure of the locus of curves in $\overline{\mathcal{M}}_g$ that admit an abelian differential of signature $κ=(k_1,...,k_{g-2})$.
For a linear subvariety $M$ of a stratum of meromorphic differentials, we investigate its closure in the multi-scale compactification constructed by Bainbridge-Chen-Gendron-Grushevsky-Möller. We prove various restrictions on the type of defining linear equations in period coordinates for $M$ near its boundary, and prove that the closure is locally a toric variety. As applications, we give a fundamentally new proof of a generalization of the cylinder deformation theorem of Wright to the case of meromorphic strata, and construct a smooth compactification of the Hurwitz space of covers of the Rie…
We describe the closure of the strata of abelian differentials with prescribed type of zeros and poles, in the projectivized Hodge bundle over the Deligne-Mumford moduli space of stable curves with marked points. We provide an explicit characterization of pointed stable differentials in the boundary of the closure, both a complex analytic proof and a flat geometric proof for smoothing the boundary differentials, and numerous examples. The main new ingredient in our description is a global residue condition arising from a full order on the dual graph of a stable curve.
Convex iso-Delaunay regions in strata of translation surfaces
本文证明平移曲面分层中，具有三角形匹配对合的三角剖分对应的 iso-Delaunay 区域在角坐标下是凸的，并由此推出超椭圆分层分量中所有此类区域均凸，推广了平面曲面理论的两个方向。
Let M be a translation surface. We show that certain deformations of M supported on the set of all cylinders in a given direction remain in the GL(2,R)-orbit closure of M. Applications are given concerning complete periodicity, field of definition, and the number of of parallel cylinders which may be found on a translation surface in a given orbit closure. The proof uses Eskin-Mirzakhani-Mohammadi's recent theorem on orbit closures of translation surfaces, as well as results of Minsky-Weiss and Smillie-Weiss on the dynamics of horocycle flow.
Weaving Geodesics and New Phenomena in Horocyclic Dynamics
本文研究了双曲面中霍罗 cycles 的动力学，首次提出了非平凡的最小轨道闭包和奇异测度的例子，揭示了霍罗 cycles 动力学中的新现象。
This article is a first step towards the understanding of the dynamics of the horocycle flow on foliated manifolds by hyperbolic surfaces. This is motivated by a question formulated by M. Martinez and A. Verjovsky on the minimality of this flow assuming that the natural affine foliation is minimal too. We have tried to offer a simple presentation, which allows us to update and shed light on the classical theorem proved by G. A. Hedlund in 1936 on the minimality of the horocycle flow on compact hyperbolic surfaces. Firstly, we extend this result to the product of PSL(2,R) and a Lie group G, whi…
We show that all GL(2,R) equivariant point markings over orbit closures of translation surfaces arise from branched covering constructions and periodic points, completely classify such point markings over strata of quadratic differentials, and give applications to the finite blocking problem.
Translation surfaces can be defined in an elementary way via polygons, and arise naturally in in the study of various basic dynamical systems. They can also be defined as Abelian differentials on Riemann surfaces, and have moduli spaces called strata that are related to the moduli space of Riemann surfaces. There is a GL(2,R) action on each stratum, and to solve most problems about a translation surface one must first know the closure of its orbit under this action. Furthermore, these orbit closures are of fundamental interest in their own right, and are now known to be algebraic varieties tha…
We study the topological dynamics of the horocycle flow $h_\mathbb{R}$ on a geometrically infinite hyperbolic surface S. Let u be a non-periodic vector for $h_\mathbb{R}$ in T^1 S. Suppose that the half-geodesic $u(\mathbb{R}^+)$ is almost minimizing and that the injectivity radius along $u(\mathbb{R}^+)$ has a finite inferior limit $Inj(u(\mathbb{R}^+))$. We prove that the closure of $h_\mathbb{R} u$ meets the geodesic orbit along un unbounded sequence of points $g_{t_n} u$. Moreover, if $Inj(u(\mathbb{R}^+)) = 0$, the whole half-orbit $g_{\mathbb{R}^+} u$ is contained in $h_\mathbb{R} u$. Wh…
Hurwitz-Hecke Invariant Subvarieties
本文提出一种基于纯组合输入和 Hurwitz 空间理论在平移曲面分层中构造仿射不变子簇的方法，并证明其可用于构造 Bouw-Moeller Teichmueller 曲线及七个 Eskin-McMullen-Mukamel-Wright 秩二轨道闭包。
Horospherical dynamics in invariant subvarieties
本文研究平移曲面模空间不变子簇上的球面叶状结构，定义叶片体积形式与球面测度，证明自然测度及 Masur–Veech 测度的唯一性，并刻画全部叶片闭包。
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