Weil-Petersson度规的动力学性质

Blaschke积空间的解析理论：同时统一化与压力度量

Analytic Theory on the Space of Blaschke Products: Simultaneous Uniformization and Weil-Petersson Metric

本文研究Blaschke积模空间的复结构及压力度量，证明了同时统一化定理并展示了压力半范数在超吸引子外的非退化性。

Cusp excursions of random geodesics in Weil-Petersson type metrics

We analyse cusp excursions of random geodesics for Weil--Petersson type incomplete metrics on orientable surfaces of finite type: in particular, we give bounds for maximal excursions. We also give similar bounds for cusp excursions of random Weil--Petersson geodesics on non-exceptional moduli spaces of Riemann surfaces conditional on the assumption that the Weil--Petersson flow is polynomially mixing. Moreover, we explain how our methods can be adapted to understand almost greasing collisions of typical trajectories in certain slowly mixing billiards.

Weil-Petersson测地流的混合速率I：非例外模空间中无快速混合

Rates of mixing for the Weil-Petersson geodesic flow I: no rapid mixing in non-exceptional moduli spaces

我们证明了黎曼曲面的非例外（高维）模空间上的Weil-Petersson流的混合率至多是多项式。

Limit sets of Weil-Petersson geodesics with nonminimal ending laminations

In this paper we construct examples of Weil-Petersson geodesics with nonminimal ending laminations which have 1-dimensional limit sets in the Thurston compactification of Teichmüller space.

Riemann surfaces模空间中Weil-Petersson测地线的性质

Prescribing the behavior of Weil-Petersson geodesics in the moduli space of Riemann surfaces

我们研究了具有窄端不变量的Weil-Petersson（WP）测地线，发展了控制长度函数和扭曲参数的技术，并规定了它们在黎曼曲面的模空间中的行程。这类测地线足够丰富，可以提供模空间的薄部分中的闭合WP测地线的例子，以及具有最小填充结束层的发散WP测地线射线。一些成分的独立利益如下：强度版本的沃尔珀特的测地线极限定理证明在第4节。第5节证明了裤子图中窄对部分标记或叠层之间层次分解路径的稳定性。根据第7节所述的次表面系数，对叠层进行一种符号编码。

Weil-Petersson测地流的拓扑动力学

Topological dynamics of the Weil-Petersson geodesic flow

证明了二维双曲结构模空间的Weil Petersson测地流的拓扑传递性。我们的证明遵循一种新的方法，利用奇异单位切向量的密度，尖点的几何形状和负曲率的凸性。我们还表明，韦伊彼得森测地流有：马蹄铁，不变集与积极的拓扑熵，并有无限多的双曲闭测地线，其数量呈指数增长的长度。此外，我们注意到体积熵是无穷大的。

Teichmueller流和Weil-Petersson流

Teichmueller flow and Weil-Petersson flow

对于非例外定向曲面S，设Q（S）是面积为1的二次微分的模空间。证明了Q（S）存在一个Borel子集E，它在Teichmueller流F^t下不变，且对每个不变Borel概率测度都是满测度，并且F^t对E的限制在Weil-Petersson流中存在可测共轭.这种共轭性导致F^t的不变Borel概率测度空间连续注入到Weil-Petersson流的不变Borel概率测度空间中。映射H不是满射的，但它的像包含勒贝格刘维尔测度。

The geometry of the Weil-Petersson metric in complex dynamics

In this work, we study an analogue of the Weil-Petersson metric on the space of Blaschke products of degree 2 proposed by McMullen. Via the Bers embedding, one may view the Weil-Petersson metric as a metric on the main cardioid of the Mandelbrot set. We prove that the metric completion attaches the geometrically finite parameters from the Euclidean boundary of the main cardioid and conjecture that this is the entire completion. For the upper bound, we estimate the intersection of a circle $S_r = \{z : |z| = r\}$, $r \approx 1$, with an invariant subset $\mathcal G \subset \mathbb{D}$ called a …

Weil--Petersson geodesics on the modular surface

We consider the Weil--Petersson (WP) metric on the modular surface. We lift WP geodesics to the universal cover of the modular surface and analyse geometric properties of the lifts as paths in the hyperbolic metric on the universal cover. For any pair of distinct points in the thick part of the universal cover, we prove that the WP and hyperbolic geodesic segments that connect the pair, fellow-travel in the thick part and all deviations between these segments arise during cusp excursions. Furthermore, we give a quantitative analysis of the deviation during an excursion. We leverage the fellow …

Rates of mixing for the Weil-Petersson geodesic flow II: exponential mixing in exceptional moduli spaces

We establish exponential mixing for the geodesic flow $φ_t\colon T^1S\to T^1S$ of an incomplete, negatively curved surface $S$ with cusp-like singularities of a prescribed order. As a consequence, we obtain that the Weil-Petersson flows for the moduli spaces ${\mathcal M}_{1,1}$ and ${\mathcal M}_{0,4}$ are exponentially mixing, in sharp contrast to the flows for ${\mathcal M}_{g,n}$ with $3g-3+n>1$, which fail to be rapidly mixing. In the proof, we present a new method of analyzing invariant foliations for hyperbolic flows with singularities, based on changing the Riemannian metric on the pha…

Lecture notes on the dynamics of the Weil-Petersson flow

This text grew out of some lecture notes prepared by the author in the occasion of a series of three lectures during the workshop "Young mathematicians in dynamical systems" organized by Francoise Dal'bo, Louis Funar, Boris Hasselblatt and Barbara Schapira in November 2013 at CIRM, Marseille, France. The three lectures at the origin of this text were part of a minicourse by Keith Burns, Boris Hasselblatt and the author around the recent theorem of Burns-Masur-Wilkinson on the ergodicity of the Weil-Petersson (WP) geodesic flow. The goal of these notes is the same of the author's lectures: we c…

拟fuchsian空间的豪斯多夫维数与Weil-Petersson扩张

Hausdorff dimension and the Weil-Petersson extension to quasifuchsian space

考虑拟fuchsian空间上的一个自然非负2-形式G，它推广了Teichmuller空间上的Weil-Petersson度量.我们完整地描述了G的正定轨迹，证明了它是关于拟fuchsian空间fuchsian对角线的正定度量，并且在fuchsian对角线的“纯弯曲”切向量上仅为零.我们表明，G等于从动力学的压力度量的拉回。利用G的性质证明了在拟fuchsian空间上的豪斯多夫维数函数的任一临界点处，豪斯多夫维数函数的Hessian在切空间的至少一个半维子空间上是正定的.特别地，这意味着豪斯多夫维数在拟fuchsian空间上没有局部极大值。

Pressure type metrics on spaces of metric graphs

In this note, we consider two Riemannian metrics on a moduli space of metric graphs. Each of them could be thought of as an analogue of the Weil-Petersson metric on the moduli space of metric graphs. We discuss and compare geometric features of these two metrics with the "classic" Weil-Petersson metric in Teichmüller theory. This paper is motivated by Pollicott and Sharp's earlier work.