Ursula Hamenstaedt
Let S be a non-exceptional oriented surface of finite type. We classify all Radon measures on the space of measured geodesic laminations for S which are invariant under the mapping class group.
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Invariant measures on the space of measured laminations for subgroups of mapping class group
本文研究曲面映射类群子群在测度叶状空间上的不变Radon测度，通过分类回归叶状空间上的测度，证明发散型子群的唯一遍历性并推广至凸余紧子群，方法几何且适用于多种度量空间。
We prove a compactness theorem for embedded measured hyperbolic Riemann surface laminations in a compact almost complex manifold $(X, J)$. To prove compactness result, we show that there is a suitable topology on the space of measured Riemann surface laminations induced by Levy-Prokhorov metric. As an application of the compactness theorem, we show that given a biholomorphism of $ϕ$ of a closed complex manifold $X$, some power $ϕ^k $ ($k>0$) fixes a measured Riemann surface lamination in $X$.
Let $\mathcal{L}$ be a measured geodesic lamination on a complete hyperbolic surface of finite area. Assuming $\mathcal{L}$ is not a multicurve, our main result establishes the existence of a geodesic ray which has finite intersection number with $\mathcal{L}$ but is not asymptotic to any leaf of $\mathcal{L}$ nor eventually disjoint from $\mathcal{L}$. In fact, we show that the endpoints of such rays, when lifted to the universal cover $\mathbb{H}^2$ of $X$, give an uncountable set $h\tilde{\mathcal{L}}\subset S^1$ (called the halo of $\tilde{\mathcal{L}}$), which is disjoint from the endpoin…
Shear-shape cocycles for measured laminations and ergodic theory of the earthquake flow
本文将 Mirzakhani 的地震流与 horocycle 流共轭扩展为所有层上的双射，构造出地震流的新遍历测度，并将其自然扩展为上三角子群作用，借助推广的剪切坐标完成遍历测度分类。
In the author's Ph.D., a version of the tangential LS category for foliated spaces depending on a transverse invariant measure, called the measured category, was introduced. Unfortunately, the measured category vanishes easily. When it is zero, the rate of convergence to zero of the quantity involved in the definition, by taking arbitrarily large homotopies, gives a new invariant, called the secondary measured category. Several versions of classical results are proved for the secondary measured category. It is also shown that the secondary measured category is a transverse invariant related to…
Geodesic planes in a geometrically finite end and the halo of a measured lamination
本文研究无限体积几何有限双曲3-流形凸核外测地平面的闭包分类，引入奇异顶概念，证明其存在与弯叶状结构的奇异射线相关，且各亏格均存在含奇异顶的端，亏格1的generic端含不可数多奇异顶。
Let $S$ be a connected non-orientable surface with negative Euler characteristic and of finite type. We describe the possible closures in $\mathcal M\mathcal L$ and $\mathcal P\mathcal M\mathcal L$ of the mapping class group orbits of measured laminations, projective measured laminations and points in Teichmüller space. In particular we obtain a characterization of the closure in $\mathcal M\mathcal L$ of the set of weighted two-sided curves.
Marden and Strebel established the Heights Theorem for integrable holomorphic quadratic differentials on parabolic Riemann surfaces. We extends the validity of the Heights Theorem to all surfaces whose fundamental group is of the first kind. In fact, we establish a more general result: the {\it horizontal} map which assigns to each integrable holomorphic quadratic differential a measured lamination obtained by straightening the horizontal trajectories of the quadratic differential is injective for an arbitrary Riemann surface with a conformal hyperbolic metric. This was established by Strebel …
A version of the tangential LS category is introduced for topological laminations with a transverse invariant measure. Here, we use the transverse measure of the contraction of a tangential categorical open set instead of counting this set. This new measured category is invariant by leafwise homotopy equivalences preserving the transverse measures, and the condition of being zero or positive is a transverse invariant. The usual tangential LS category is also bounded by the number of certain critical sets. It is also proved that the measured category is semicontinuous when the foliated structur…
Let $S$ be a compact, connected, oriented surface, possibly with boundary, of negative Euler characteristic. In this article we extend Lindenstrauss-Mirzakhani's and Hamenstädt's classification of locally finite mapping class group invariant ergodic measures on the space of measured laminations $\mathcal{M}\mathcal{L}(S)$ to the space of geodesic currents $\mathcal{C}(S)$, and we discuss the homogeneous case. Moreover, we extend Lindenstrauss-Mirzakhani's classification of orbit closures to $\mathcal{C}(S)$. Our argument relies on their results and on the decomposition of a current into a sum …
We first show that an earthquake of a geometrically infinite hyperbolic surface induces an asymptotically conformal change in the hyperbolic metric if and only if the measured lamination associated with the earthquake is asymptotically trivial on the surface. Then we show that the contraction along earthquake paths is continuous in the Teichmüller space of any hyperbolic surface. Finally, we show that if a measured lamination vanishes while approaching infinity at the rate higher than the distance to the boundary then it must be trivial.
On stationary actions of locally compact groups and their Radon-Nikodym cocycles
研究局部紧群的平稳作用及其Radon-Nikodym共轭函数的结构与正则性，证明平稳作用非I型、非II_∞型，并建立通用的紧Radon-Nikodym模型，同时构造实仿射群上的随机游走，其Poisson边界不满足Kaimanovich的SAT*性质。
The Radon--Nikodym topography of acyclic measured graphs
本文研究无环测度类保持Borel图的Radon-Nikodym地形，通过分析端点数量与可接受性、平滑性的关系，提出非消失端点决定图的性质。
We interpret the probabilistic notion of unimodularity for measures on the space of rooted locally finite connected graphs in terms of the theory of measured equivalence relations. It turns out that the right framework for this consists in considering quasi-invariant (rather than just invariant) measures with respect to the root moving equivalence relation. We define a natural modular cocycle of this equivalence relation, and show that unimodular measures are precisely those quasi-invariant measures whose Radon--Nikodym cocycle coincides with the modular cocycle. This embeds the notion of unim…
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